Man, Puyo Puyo used to be one of my favorites puzzle games. However, I have been outta the Puyo loop since the release of Puyo Puyo Fever...do the newer ones bring something fresh to the table, gameplay-wise ?
Puyo Puyo 15th Anniversary: Fever mode was altered so that when the timer hits 0, your fever is over right then and there (it's more exact and strict). The fever chains are kinda crummy too lol. They're mostly OK, but not so awesome for competitive play. There's also a bunch of party modes and stuff like Ice Blocks, Spinner/Flip, and Underwater, but those can't be played through wifi.
Puyo Puyo 7: Party modes are out afaik, so only legacy modes (Original Puyo Puyo, Puyo Puyo Tsu/2, Puyo Puyo Fever, Puyo Puyo Transformation, Nazo/Mission Puyo Puyo) are in it. Only Transformation, Fever, and Tsu rules can be played online in random battle and friend games (PP15th was Fever and tsu in random, and Fever tsu and OPP in friend games).
Fever mode is still the same, except if you screw up your 5 chain you get a 3 chain next. (In pp15th the minimum given chain was 5).
Transformation mode is really different, and not suited for competitive play (even though it's supposed to be...). I made a Transformation Mode mechanics guide. http://puyonexus.net/forum/viewtopic.php?f=2&t=1167