From what I noticed....
Playing PS1 games on a PS2 has some slowdowns on ingame special effects.
Games that I noticed it with were, Legend of Dragoon, FF Anthology, FF Chronicals and Legend of Legaia. There are others, but I don't really remember what they were.
In Legend of Dragoon, the Battle Transition takes way too long. What should take a few seconds takes about 6. In Legend of Legaia, the summons are animated too slowly, causing that part of the game to be longer than it should. The Sound ends a good 7 seconds before the Animation does, sometimes longer. In the FF remakes, Saving and Loading of the games takes way too long.
Anyway... Those are the types of compatibility problems that I've noticed with the PS2 playing PS1 games. Mainly, Slow downs during special effects, but nothing really gamestopping. As long as you use a Sony Memory card, that is. And not a 3rd party one.