If something is embarrassing, it is nearly always something you wish had not happened. In most cases, it involves other people who can make you feel uncomfortable or humiliated. "I was embarrassed when I forgot to zip up my pants."
If something is awkward, it is usually something that either could have been done better, or will get better with practice. Something can be awkward even when you are by yourself. "I usually feel awkward the first time I try to do something I have not done before."