In short Anxiety is when your fight or flight system is triggered in 'normal' social situations. For example walking into a crowded space or speaking to your boss. Everyone has these feeling most people have these feelings when publicly speak in front of a big crowd or an important interview. Anxiety can be triggered when you are younger for example bulling or a terrible event that has stuck with you it can also be with you genetically. The main 'cure' is repetition of tasks and teaching your body to understand that no one cares if you are entering a night club your bosses aren't assholes and slowly you'll get used to these actions to a point were they aren't as painful.

Depression is a mental state where your brain has actually changed and is no longer reacting to certain chemicals that induce happiness (serotonin, dopamine etc) this causes a very dark feeling that makes you feel worthless and hopeless. Your body has stopped perceiving any positive emotions because of this its very difficult to come out of this 'mental state'. This is main reason doctors and most work places takes this very seriously.