Phantasy Star Online was fantastic. It had some connection to the original games, but many new ideas and systems. The game was originally developed without plans to make it a Phantasy Star game and that was bolted on midway through development, which could account for why the plot was somewhat lacking. Overall though, it was like most Phantasy Star games - There's the Dark Force/Profound Evil, it's possessing people and animals and corrupting things, and you have to stop it. Usually they've explained the discrepancies by setting various games in the past. I think ultimately you're supposed to just not care and accept that ok, it's a PS game, of course you're fighting the Dark Falz.
Phantasy Star Online 2 was perhaps even better. It is a huge game, and was around in Japan for years, receiving many large updates, without a US release. Eventually Microsoft brought it to the US because they wanted to be involved with the successor, Phantasy Star Online 2: New Genesis. It was released with a rapid fire schedule where they compressed 9 years of updates into 2 years. Now, it's unfortunately not fully playable, and receiving no more new content, but is still mostly available. NGS has been doing... okay. The success of that game is mixed. It's decent though and is still receiving new content.