I'll admit, I'm a sucker for Rotel.
I've always felt they offered unparalleled beauty and design for the money.
The RB-1582 has all that I love. It's got a huge toroidal transformer, and a cache of caps that makes me drool. It's got 4 - 15,000uF 80v caps, which is plenty of juice for practically anything. Has 8 high output transistors dedicated to each channel. Using oversized heatsinks the Rotel is able to keep the heat down to a minimum, and at worst will be warm to the touch.
Comparatively the Anthem A2 has a similarly sized transformer and the same level of capacitance for each channel (can't verify voltage of caps, but the microfarad values are the same). Also, like the Rotel, the Anthem uses 8 high output transistors per channel, and again runs very cool with the use of oversized heatsinks.
IMO, they are going to operate very similarly. Both are excellent designs, but I prefer the looks of the Rotel and it's cheaper for you.
Sorry, not much for sonic differences between amplifiers like these. They are both competently designed, and intended to drive a linear signal.