Triple Strange is a non-profit, and we've seen considerable success.
We started off in 2005 as a school-based team and continued that until internal disputes from a new principal forced us away from our school and into an independent area. Changing to a non-profit was a leap, but after getting set up, things began to grow considerably. Probably the biggest benefit of being a non-profit is the ability to expand your organization. "ECG Robotics Inc.", the non-profit that everything runs through, is now housing our FRC team and four sister FTC teams within our build space. Being under your own management allows you to expand greatly and set your own rules to avoid the ridiculous restrictions set by the school boards sometimes.
tl;dr/Pros and Cons:
+Can pull members from a region rather than from a school
+Can expand teams as needed without as much hassle
+Get to avoid school regulations (Snow days, transportation laws requiring students to ride buses to events, etc.)
+Once infrastructure is set, you just need a constant flow of parent/mentor support and you're good to go.
-Takes a lot to set up infrastructure (501c3 license, tools and build space, community awareness of club)
-Takes a lot of parent/mentor support
All in all, you're able to do a lot more once you have an independent team, it just requires a lot of infrastructures. If you're thinking of branching off from a school to an independent team, contact sponsors and businesses to see if they can offer build spaces, tools, or other help.
Also just a fun fact, you can find every independent 501c3 robotics organization through GuideStar
This is our organization. If you're part of a district, feel free to look for your districts non-profit. FIRST's non-profit data is all available to look at for free.