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Building from this question, I learned that it is possible to notate a tie that is held by the pedal and not the hands. To me that's a bit strange - if the pedal were not held down in that passage, the effect would be different because the right hand is playing more notes that would not be sustained.

That brings me to the question: if I play a chord without the pedal and hold it for 3 seconds, and if I play the chord with the pedal, releasing it after I play it so that the pedal holds the notes, are the effects going to be the same?

I recognize that I can just go ahead and play this myself (I have), and personally I couldn't tell a difference. But, since there is an obvious physical difference (the pedal either is or is not being pressed), I was wondering if internal to the piano there is a difference that would make a musician choose one option over the other. Remember, I'm asking about a single chord. I recognize that there is a big difference if multiple chords or notes are played successively.

Playing a chord lifts the dampers from just the strings of the notes played. Depressing the pedal lifts the dampers from ALL the strings. A lot more resonance.

I'm amazed that you can't hear the difference! Maybe you aren't playing a real piano, but a basic-grade electronic imitation that doesn't model this important part of the piano sound?

One of the most distinctive features of the piano (as opposed to e.g. the harpsichord or the clavichord) is the sympathetic resonance. Whenever you hit one string, all the strings sympathetic to that one (*) will vibrate along with it. Usually the dampers get rid of this resonance immediately, but holding down the damper pedal lets it shine through.

Try this: hold down middle C (C4) and/or the G above it (G4), then keep holding them after the sound's died away. This keeps their dampers off so they can resonate properly.

Then, while still holding those keys, strike and release the C below middle C (C3). When you release C3, its own sound will be cut off—but the other strings will still be sounding! This is the extra sound that you get when you hold down the damper pedal, except the damper pedal gives you every sympathetic note, all at once. It's a much richer, warmer sound.

(*) How "sympathetic" two strings are comes down to how well their harmonic series overlap. Try the experiment again, but this time hold down D4 and strike C4. You won't hear much resonance, because those strings aren't very sympathetic to each other.

Using the pedal, there is sympathetic vibration from other strings that would otherwise be dampened. This leads to more of a "chorus" kind of sound. You can amplify the effect by striking a chord strongly and very shortly and right after releasing the keys pressing the pedal. While the strings of the struck keys will still contribute more to the resulting sound, the contribution of the other strings becomes quite more obvious.

On my old Acrosonic Baldwin (1947), I have played with the center pedal down to get a kind of resonance that's similar to "hall" on electronic pianos.

Of course, I have to stay at the upper end and steer clear of bass notes as those will just sustain indefinitely when center pedal is down.

Side note: What's that pedal really used for, anyway, and does it have a name?