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Because boiling a volume of water creates a much bigger volume of steam this volume increase is about 1700 times.

According to the gas laws doubling the absolute temperature of a volume of gas only doubles the volume at the same pressure, or doubles the pressure at the same volume. Doubling the absolute temperature means going from 300 K (27°C) to 600 K (327°C). So to get working pressures you need to heat to extremely high temperatures.

As an additional benefit the pressure of steam is related to the temperature of the boiler. This means that if you can maintain temperature there will be no pressure drop until you run out of water to boil. With pressurized gas the pressure will drop as you use gas unless you increase the temperature even more.

Hot air engines are feasible, and have a 200 year history behind them, starting in 1816 with the Reverend Robert Stirling.

The other answers are largely correct : steam offers more energy per unit volume, but theoretically lower achievable efficiency while the Stirling Cycle can (in theory) match the ideal efficiency of the Carnot Cycle, using a regenerator to store teh bulk of the heat removed in the cooling phase, and return it to the air in the heating phase.

However there are practical difficulties transferring heat into and out of the working fluid fast enough, while minimising flow losses; good heat exchangers keep gas in close contact with the heat exchange surfaces, maximising heat exchange rates, but also maximising frictional losses.

Steam wins here; the bulk of the heat exchange is in the liquid phase, which is much more effective (compare water cooling with air cooling).

Hot air engines surface every few decades; Philips did some impressive work achieving close to 40% thermal efficiency between the 1950s and 1970s, and this work lives on in some Swedish submarines where its relative silence is an advantage.

They also make cute toys capable of running off modest heat sources - like the warmth of your hand.

There are really countless possible answers to this. I'd like to add one that hasn't been made clear yet.

If you are doing a closed cycle, where you aren't letting steam/gas out into the atmosphere, at some point you need to pump the fluid back into the boiler to gain energy again. Pumping gas is very difficult to do. Pumping liquid water is very easy. So in the case of closed systems, it's terribly convenient that we can condense the steam back into water, and then move it mechanically back into the boiler to create new steam.

While toy steam engines often require that one pre-fill the boiler, and then only operate until the boiler is nearly empty, most practical steam engines require that every gram of steam which leaves the boiler be replaced by a gram of water during operation. The amount of work required to pump a gram of cold air into an engine is slightly less than the amount of work that can be obtained from a gram of hot air that leaves, but not by a whole lot. The amount of work required to pump in a grab of water, however, is far lower.

Further, if a steam engine includes a condenser, that can be used to reduce the pressure downstream of the pistons to below atmospheric. This partial vacuum contributed substantially to the power output and efficiency of early steam engines, but becomes less significant as working pressures increase. Steam locomotives don't bother with a condenser because getting high power outputs requires using higher working pressures, and the relative usefulness of the condenser for power generation goes down as working pressures go up. The general lack of condensers is perhaps somewhat curious given that maintenance costs are strongly related to the total quantity of contaminants brought into an engine, and recirculating water would reduce the quantity of new water (and thus contaminants) it would be necessary to add.

Do a web search on "Rankine cycle".

The real key is that in a practical heat engine, you need to pump in cold working fluid, heat it, let it do work (and cool) as it expands, and then repeat.

If you ignore losses, the energy to pump a working fluid up by a given pressure difference is proportional to the volume times that pressure difference. A mole of liquid water is much smaller than a mole of gas. So a heat engine that pumps liquid water (or any other liquid that will subsequently be boiled) has a significant advantage over a heat engine that does the job with a gas.

For Air:

Consider the ideal gas law PV = nRT rearranged to be V = (nRT)/P.

n and R are considered constant, and for our purposes, P is also constant. Thus, to increase the volume, (i.e. create a flow of volume across the turbine to power it), T has to be increased. From this equation, V is proportional to T. so for every multiplicative increase of T you get the same increase in V.

To get an increase of 1000x in volume, T has to be increased by 1000x.

For Steam:

Now when we consider steam, it is a much different story. This equation is no longer valid. I think it would be best to look at the density of water (liquid) vs water (steam). liquid water has a density of 1000 kg/m^3 and we will say steam is 1 kg/m^3. There are variations possible here based on pressure and temperature etc, but we will go with what I listed.

Since density is mass/volume, we can see from these numbers that when water is boiled off, its volume increases 1000X!

Since water can be liquid < 100 °C and Steam > 100 °C, the temperature would only have to be increased by a few degrees to get 1000x expansion!

An additional note here too, is that the steam loses a lot of energy when it is running a turbine, and is condensed back to a liquid state. Not all of it is, but a lot of it is (Conservation of energy). This condensed steam may now be used again and be boiled off to run the turbine more.

For air, this phenomena does not exist and the air has to be compressed/cooled down all over again.

Conclusion:

There is a lot more going on here, but this is the main principle here. Air can be used, it just shouldn't be used. Water is just more efficient due to the phase change -> density/volume change.

As said, boiling water creates much more gas. To make a useful engine for the kind of applications that steam engines are used in, you want to build lots of pressure to be able to move heavy stuff, e.g. a train for which the locomotive alone may be 10 Mg (megagrams, "tonnes") or more in mass.

If we look at the ideal gas law, written to emphasize pressure, i.e.

$$P = frac{nRT}{V}$$

we can see there are three ways to increase the pressure: increase the temperature $T$, which is what you're suggesting, de crease the volume $V$, which you can't do here because it's a fixed-size chamber, or you can increase the number $n$ of gas particles (molecules) (technically, bulk number in moles rather than direct number, for that, replace $nR$ with $N k_B$ above).

(Another, hypothetical way, as you can see, would be to somehow change $R$ ... but that's called "magic" and "changing the laws of physics". Sadly, we don't have that kind of power 😁)

Liquid water contains about 1000 times the number of particles per unit volume than gaseous water does and - more importantly - air. And it generates all that gas simply by boiling it. Thus, with a volume of material that easily fits in the chamber, you can create from it up to 1000x more gas than a similar volume of air would give you and thus 1000x the pressure at any given temperature.

Now, the gas law certainly also works just as well for air, too, and thus you could also, with air, increase $n$ and thus also $P$ in the same way. The problem with that is what it takes to do that. Air is, under ordinary conditions, already a gas: and thus to get more $n$ into the chamber, you have to pump in gas and increase the pressure. That means you are now compressing air ... and congrats, you have just re-invented the compressed-air engine, only now with an added heater. Better to just burn all that fuel to run the compressor and compress the air from the get-go, mount the tank, and leave out the pyrotechnics.

The alternative would be to use air that is condensed into a liquid or solid form. You can do that - it's called liquid nitrogen (well, okay, that's 75% of air, not 100%). The problem with that is you have to expend energy in cooling it down, now, to make it liquid, since room temp - 295 K (or 300 K if you're in China or many other parts of the world) - is way above the boiling point of nitrogen, 77 K. Thus, it's much more efficient to just heat up and boil something that is already liquid at room temperature.

Now on that last note, water isn't the only option in theory - another would be alcohol, but alcohol is flammable: you heat that stuff up in your engine and unless it were a completely inert atmosphere therein, it would catch on fire and burn all at once and you'd have a bomb, not an engine. Moreover, even if there is an inert atmosphere, the alcohol may pyrolyze (decompose, i.e. the molecules fall apart) before it reaches a good working temperature, and while in theory you could actually say this is good because it produces more gas by breaking big gas molecules into little ones and hence rising $n$ even further, decomposing the material consumes even more energy. Moreover, the breakdown products of this include gases that you cannot re-liquefy at room temperature and hence we're back to the compressed air engine. Even moreso, they include water ($mathrm{H_2O}$) and thus we're also back to the steam engine. Finally, the $n$ of alcohol yielded per unit volume is smaller than water owing to the larger sizes of its molecules.

So at the end of the day, you might as well tank it up with water. It is liquid at room temp, boils at a relatively low temp, has lots of molecules to give up, is extremely common and cheap (esp. if you get it from the ocean and not use precious freshwater reserves since you don't need to drink any of this, moreover, distillation is trivial since effectively your engine "distills" it many times and thus doing a pre-run outside is trivial added expense), and anything else you might use will stand a strong likelihood of turning into it anyways, and insofar as it is concerned, it itself does not pyrolyze until around 3000 K or so, well above the melting point of iron, 1811 K and thus above the point where your engine turns into a lava puddle.

Water is where it's at. In fact, I might hazard that it's the optimal material for this purpose: generally speaking, more complex molecules fall apart more easily (more complex things, break more easily in general, as a universal principle), and of simple molecules (e.g. 3 atoms or so) all are either gases at room temp and/or dangerous reactive substances (e.g. hydrogen chloride).

Now, if you were building an engine to operate on an extraterrestrial body like Titan or even Pluto, you'd have the option of using solid or liquid nitrogen or methane/ethane as it's cold enough these are now available "for free" like how water is, without any special chilling steps. While the problem of decomposition temp for the latter still technically remains, you have to remember engines, and the gas laws with regard to temperature, operate on ratios of expansion and with liquid methane being boiled you're starting at 111 K instead of 373 K, thus going from just over 111 K to 333 K, still below the decomp. point of methane, gives you roughly the same expansion as going from 373 K to 1119 K, as may be done in a real steam engine.

The trouble is, now you're going to be more worried about fuel/oxidizer combos, and you'll probably thus want to just use them more directly in an internal combustion-like setup. The whole point of steam engines was that you could use felled wood and/or coal, together with atmospheric oxidizer. Nothing like that on these worlds.