What are the differences between these PWM controllers, specifically related to driving servos?
RC servos need an input that goes to +5V for a precise time. This pulse length is the position command (1ms = move full left, 1.5ms = move to center, 2ms = move full right). Then the input needs to go to 0V for some period of time (this period usually doesn't matter much). Any chip that can do that should work.

I think I see 2 reasons why the TLC5947 isn't good for servos:


If you look at "Figure 3. OUT0 Through OUT23" in the TLC5947 datasheet, you can see that it has an "open-collector" output. This means that in one state it connects to ground, and in the other state it simply disconnects from everything. This is fine for switching LEDs on and off. But servos require a "push-pull" logic output: in one state it connects to ground, and in the other state it connects to +5V. You can use a pullup resistor to achieve this result with an "open-collector" output, but now you're adding an extra component.
I can't figure out what the pulse frequency is. The TLC5947 might not offer precise control of pulse length, which you need for servos as discussed above. I might just be missing it though.