You answered your own question with 'varying amounts' I think? assuming there's a dry/wet on the compressor anyway..
They're essentially the same thing, a way of summing multiple audio signals
DAWs all have their own system, if you look at an actual mixer it makes more sense, you have the aux sends, usually with pre/post fade and send level controls, you can patch in inserts if you want to send the whole signal to some external processing and you have the sub-groups that you can use to bus a sub mix (drums etc)
BUS is just a signal path to route audio (1 or more inputs) to a specified output, which is what a group does in Ableton, but in logic you can control the amount being sent to the bus, which makes it an aux send as well so it varies between DAWs
The real difference is between busses (2 or more inputs at output stage of signal) and inserts (1 input at input stage)