H-bridge problem with TIP122.
The upper TIP122's are wired as emitter-followers, so their output voltage can never be higher than their input (base) voltage. (In fact it will be ~ 2 x 0.6V lower.)

When you fed the inputs with the full Vcc that was not much of a problem, but when you feed it with the 5V Arduino outputs you will get no more than 5V from the outputs.

One solution is to put a level shifter between the Arduino outputs and the H-bridge. But it will still be a crappy H-bridge. (Do you realize what will happen when you make BOTH inputs high?)