There are not a lot of differences between all the arduino based flight boards out there to be honest.
Any avr based arduino (uno, leonardo, mega) have the same AVR processor, the only computational differences are RAM and flash memory sizes. They are all capable of running a quad, if bounded somewhat on how fancy they can get by their slow computations. AVR is intended for IO, not processing power after all.
A more powerful arm processor would aleviate many problems by running a RTOS and having much quicker calculations, but then you mave more IO problems because the ARM arduinos are 3.3v while the RC electronics normally used are 5v. A lot of the newer boards I see running quads are ARM. They are generally not arduino compatible though.
Anyway, the arduino mega (or any board with an atmega2560) is the best arduino to use because of its extra memory, but if you are careful any avr with 2kb of ram or more should be safe. Sensor wise it absolutely much have an accelerometer and gyroscope. Magnetometer and barometer are good to have but unnecessesary. Usually this means it will have an MPU6000 or MPU6050 onboard.
You should look for boards that are supported by MultiWii, aeroquad, or ardupilot. They will have what you need. ready to fly quads and hobbyking are where I would look. Then you can upload some open source code to run everything.
Make sure you know how much work goes into writing the software to run a quad yourself if you are planning on doing that. It is very difficult, with
the precise timing involved,
advanced mathematics (calculus, linear algebra, lots of 3d geometry),
multiple IO devices (radio in, serial in/out, accel,gyro,mag,baro, 4 motors),
and limited resources. Then, on AVR at least, you don't have threads. If any code stops the (sensor read -> flight calculation -> output update) process from happening at ~100Hz it will probably crash.