Primarily, the number of channels - one, two or three. The A500x1 runs in bridged mode and while it will drive a 4 ohm speakers, 8 ohm speakers are recommended.

Generally, the A250x3 is recommended for tri-amped applications, the A250x2 for bi-amped applications, and the A500x1 or A250x2 for full-range passive speakers. If in doubt, ask DEQX for a recommendation for your system and speakers.

The A250x3 is compatible with the HDP-4 and HDP-5. The A250x2 and A500x1 are compatible with the HDP-4, HDP-5, PreMATE and PreMATE Plus.

No, the PreMATE (and HDP-4) do not have a 12V trigger output. 

Specifications

Key specifications (all amplifiers)
Frequency response 10 Hz – 50 kHz (+0/−3 dB)
Distortion < 0.0018% (< 10Hz – 20kHz AES17)
Dampling factor > 1000 (f < 20 kHz, Load = 4Ω)
Power output
A500x1 A250x2, A250X3 Low/Mid A250x3 High
Maximum power, 4 ohms*1 400 WRMS 250 WRMS 100 WRMS
Maximum power, 8 ohms*1 500 WRMS 200 WRMS 100 WRMS
*1. 1 kHz, THD=1%, both channels driven.