Bloom is basically an effect simulating how light "bleeds" onto an object while HDR simulates how the human eye reacts to changes in the amount of light received. And HDR also features a bigger "scale" of light, which means there will be a lot of shades between really dark and really bright. Usually HDR also has great quality reflections as a feature. Usually they're both used in order to get the best visuals.
HDR is a lot mroe CPu intensive and creates a more global effect, meaning its visible in most situations, Bloom is mroe of an effect put on individual objects.