Bowdlerization: The original game was renamed "Jet Grind Radio" in America due to there being a band named Jet Set Satellite, and they feared that people would associate this game with them. The radio station is still referred to as "Jet Set Radio" in-game however, and the only changes are the title screen and the graffiti saying "To Jet Set Radio" in the intro is, for some reason, changed to "To Jet Grind Radio". This conflict was long gone by the time that JSRF and the HD re-release of the original came out, although the GBA port still retains the "Grind" title.
Some songs in both games had certain lyrics cut in order for the games to retain a Teen rating. For example, in "Birthday Cake", the original version has a verse that goes: "It's moldy mom, isn't it? // I DON'T GIVE A FLYINGFUCKTHOUGH!!!" The game's version cuts right to the chorus after the "It's morning, mom..." lyric. Another example, "I'm Not a Model" had a segment where a woman goes into disturbing detail on giving oral sex. This part was cut in the game version for obvious reasons.
Urban Legend of Zelda: With this many unlockable characters, it seemed natural at the time that Professor K must be playable — despite his incompatible character model. The same goes for the elusive Coin, too.
Professor K is an unlockable character in the GBA version though.
For a long time, people were unsure if Future was originally meant to be for the Dreamcast, seeing how the game has barely any use of the right analog stick, and it would make sense for them shift a new JSR to a new console after the failure of the Dreamcast, and the Xbox just happened to have similar architecture. Sometimes a rumor would pop up of it being confirmed that it started on the DC, and even a few times has this page had "it was originally for the Dreamcast" as an item. However, this Tweet by the game's composer finally puts the issue to rest: Future was always meant for the Xbox.
Keep Circulating the Tapes: Many songs featured in the games (everything by the Latch Brothers, Guitar Vader, and most single songs contributed by artists) have yet to see any form of re-release and in some cases, the original albums(if they were on albums) are difficult to find, leading to the games to being the best and cheapest way to hear them.
When the first game was nearing its American release, a Sampler CD was handed out from select stores. This CD included songs that were going to be added to the American version, but never were.
In an early build of JSRF, Yoyo had a smaller health capacity and might have been able to hold more than 30 spray cans (the spray can icon doesn't turn green on the footage of this build). Footage can be seen on this Xbox commercial at 0:09.
There is a rumor going around that there were three characters Dummied Out in JSRF, named Wave, Numa, and Check. Their files can only be found in the Japanese version, their files were removed on the American and European discs. No one has been able to confirm this though.
Is there a possibility that two of those characters ended up being Cube and Combo, who didn't appear in the Japanese version?
A new JSR game was proposed by Headstrong Games for the Wii, its concept art has been circulated around the internet. Sega reportedly shot down the idea, and Headstrong ended up making a new House of the Dead game instead. Fans were not happy.
Woolseyism/Dub Name Change: Several characters went through various name changes between the first game and Future, sometimes not even related to their Japanese name. Two characters in particular went through three names: Bis/Mew/Rhyth, and Sugar/Pirahna/Boogie.
One of Us: Hideki Naganuma(both game's main composers) is a huge fan of the games and has been rallying fans on Twitter to get interest up for a new game. He's also apparently a big wrestling and anime fan and also enjoys Space Channel 5.