I've got a Logitech Dual Action Gamepad. Seems very similar to the wireless gamepad you have, so i would assume they are mostly the same. It's basically a perfect replication of the PS2/PS3 controller. It even has clickable button analog sticks (L3/R3). The Xbox 360 controller has the same amount of buttons as it (except the Xbox home button). I would highly recommend the dual action at least, i've had it for quite a while and it's very sturdy and all the buttons, dpad, and analog sticks feel great. The dpad is really impressive and responsive.
I have to assume that your pick is mostly the same. If it's anything like the dual action, it's a really solid gamepad. Looks like your wireless one has rumble (mine doesn't, i never liked rumble much) AND a home button (it's very close to a real Xbox 360 controller in terms of function). If you run into a case where a real xbox 360 controller is a requirement (Mini Ninjas was the only one for me), there's an Xbox 360 controller emulator you can find floating around that forces games to work with your picked controller. Worked for me. Or you could give xpadder a try for other games you're having trouble getting to work with a controller (i was able to force keyboard-only game Beyond Good and Evil to work with my gamepad using xpadder).