Can I have different audio come out of my computer and monitor? I previously had my monitor set up through a VGA cable and I was able to play music through an audio jack on my monitor — All while listening to something else through my headphones plugged into my computer. Now that I switched from VGA to HDMI, it doesn't seem to do the same thing. 

Is this possible with an HDMI cable? Or is this only a VGA thing?
HDMI (unlike VGA) combines audio and visual in a single cable — VGA just does visual. 

Your monitor was using the audio jack as an input, and playing sound from the source connected there (music player, etc.). Your computer was using its audio jack as an output, playing music to your headphones.

Now that you have an HDMI cable, your monitor most likely uses its audio jack as output, because it already has an input for sound — the HDMI cable. My guess would be that if you plugged in headphones to your monitor, you would hear the sound coming from your computer.

There is a way to have the monitor and computer speakers play the same output (Audio MIDI Setup), but I don't think you have have two different outputs.