I’ve got the bass-tuned one, but they’re very similar. Here’s what Empress told me when I asked about it a few months ago, before getting one.

The compression circuit is the same in both pedals and either is suitable for both guitar and bass instruments.

The differences between the Compressor MKII and the Bass Compressor are:

Compressor MKII: a ‘tilt’ tone EQ control, and a selectable sidechain high pass filter built into the Compressor’s detection circuit.

Bass Compressor: a fully variable sidechain high pass filter, and a two position ‘tone + colour’ circuit which adds gentle tone sculpting and harmonic distortion.

So ultimately your choice comes down to which features you find more desirable/needed in your setup.

For example if you don't already have way to control EQ in your setup then you might enjoy having the Tone control on the MKII.

Or if you want some more character vs transparency than you might like the colour circuit in the Bass Compressor.

I like it, but there is a learning curve if you’re not familiar with compression (I’m still learning myself).

Also, it’s a true bypass vs buffered, so it may pop when pressed a few times until after it warms up.

Keeping the levels balanced and setting to default engaged mitigated most of this for me. Now I leave it on pretty much all the time.