Oh hey, I remembered my password and username!
I actually got my start in robots doing combat robots (battlebots, lower case B) in ~2001-2002 after watching the show... FIRST came later, where I founded my high school's team after already having competed in small local events for a few years.
Overall, I have found battlebots-with-a-lower-case-B to be a more freeform experience. The cost can be substantially lower to start, there are less rules so you can experiment and do crazy hackish things you wouldn't do in FIRST or Vex.
Since leaving high school, I've mentored both FIRST teams and students building combat bots for local events. Honestly, the combat world has never been as league-organized as FIRST, but maybe that's a good thing. To me it means people can pick and choose how they want to experience building and engineering.
(BTW, the "drama" on the show was entirely builder-side. Ray Billings the Evil Villain is the most chill guy in real life, for instance. We all amped it up a bit for the presentation. So long as it's all intiated by the builders I think it's okay.)
... (Also, posting in Bite Force thread as Overhaul captain
)