Are there any differences between them, And Is there an online mode on the PS3 version?
Monster Hunter Tri isn't available on the PS3, it was a Wii exclusive. An expanded version (Monster Hunter 3 Ultimate) is currently available for the WiiU and 3DS.
Are you talking about Monster Hunter Portable 3rd? Both versions of that game (PS3 & PSP) were only released in Japan. The PS3 version is an HD remake and re-release of the PSP game.
Monster Hunter Portable 3rd is the third game to be released in the series for handheld systems. Monster Hunter Tri is the third game in the series for consoles. This excludes expanded versions like Monster Hunter Freedom Unite or Monster Hunter G. This is why both games use some variant of 3 in their naming systems.