Output to bus is directly affected by the fader level on a 1:1 basis and is after all processing.
Aux send has it’s own level as well on top of this to modify even farther and can be placed before the fader (Or anywhere in the chain). Aux send is more flexible, but outputting to the bus is simpler.
It becomes more a workflow terminology and workflow. If I want to process things together, outputting to a bus becomes the better way so I don’t need to manage the level except through the channel fader, and don’t have to worry about where it goes in the processing chain, or if I add a new processor remembering to move that processor before the send. I use aux sends to handle additional sends other than this, such as sends to reverbs, sends for external mixes such as for headphones, etc.
Now one thing to note, is that there is a difference to applying compression to a bus with multiple channels feeding the input of the bus like you describe, and applying compression individually to each channel. Make sure you think t his through and decide what it is you actually want, it is rare I have the same compression on snare, kick, toms, etc. and often times I might have individual compressors and a bus compressor, depending on the style of music I am mixing.