I'll preface this by saying that everything I know about the Challenge comes from reading about it (it's really too far from California for me to go, even though I'd like to), and I've done a total of one LeMons race.
The big difference is that the Challenge is autox, drag racing, and concours, while LeMons and ChumpCar are wheel-to-wheel endurance road racing. The safety equipment required for autox is cheap, as is drag racing at least at the power levels you're likely to be encountering to begin with. OTOH, with LeMons & Chumpcar you need a cage, harnesses, fire system (for Chump), nomex driver's suit, gloves, shoes, HANS (OK, not required, but a really good idea), etc. If you're running in the summer I'd add a cool shirt system to the list of not-mandatory-but-a-really-good-idea stuff as well.
And yes, a LeMons or ChumpCar event is 16+ hours of driving hard, whereas the Challenge is, what under 10 minutes total? Mechanical failures (not just motors, but suspension, brakes, etc) are a major concern for endurance racing. Bringing an underprepared car with a lack of spares is a recipe for spending most of the event wrenching on the car in the paddock and driving the tow vehicle all over town looking for parts rather than driving the race car on the track.
Do you have a tow rig? It's possible to street drive a Challenge car to the event, but it's really not possible to do that with a LeMons car. (Well, I'm sure some people have done it, but street driving a caged car without helmet is a significant safety concern).
The downside to the Challenge event is that, AFAIK, there's only one such event a year and it's in Florida, so if you don't live in the south it's a pretty significant travel commitment. You can always run a Challenge car in your own local autox events, but that's not really the same thing.