There is a difference in the sound: OPL2 is a 9-channel mono chip, while OPL3 is a 18-channel stereo chip. Thus, sound cards containing two OPL2 chips (SB Pro 1.0, Pro Audio Spectrum and few others) have to be programmed in a somewhat different way from OPL3 cards.
If you try to play OPL3 song on a dual OPL2 card, you will end up with mono sound. If you try to play dual OPL2 song on OPL3 - the result, I believe, depends on the engine delivering song data to the hardware. For AIL 2.0, right channel is missing, leaving us with only the left and center channel.
Some websites I have found claim that this causes no difference in the sound.
This is not true overall, but might be true in some specific cases.
OPL2 and OPL3 seem to be different chips entirely.
Not entirely: OPL3 is fully backwards compatible with OPL2. But it doesn't make an OPL3 card fully backwards compatible with _dual_ OPL2 cards.
I have some games that I want to record that were designed specifically for OPL2. If I get an OPL3 SBPro 2.0, it really will not sound any different at all?
If the games were designed for a single OPL2, you will not hear any differences related to the sound chip. However, there might be some sound quality differences caused by different sound path design and quality. E.g. Sound Blaster 2.0 is a very noisy card that is prone to coil whine and even mouse movements cause noise. Pro Audio Spectrum 16 is, on the other hand, very quiet card and its sound path is isolated from extraneous noises. So, if you move from SB 2.0 to a PAS16, there will be no differences caused by OPL3 itself, but there will be ones caused by overall build quality of the involved cards.
If the games were designed for a dual OPL2 card, you won't get right sound out of an OPL3 card.
I really would like to verify this beforehand.
This isn't hard, just play with DosBox, switching its sound engine between 'sb2' (single OPL2), 'sbpro1' (dual OPL2) and 'sbpro2' (OPL3) should show you the difference for any songs you want.
Hardware comparisons and game system requirements: https://technical.city