If something is embarrassing, it is nearly always something you wish had not happened. It makes you feel uncomfortable and sometimes humiliated. It almost always involves other people. "Knowing that I made a serious mistake and others found out about it, was very embarrassing." 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." Something awkward can be embarrassing when others are around, but something embarrassing almost always makes you feel awkward. Being embarrassed is always a negative feeling. Feeling awkward is not always negative. I felt awkward the first time I had a session to teach English to a student from another country, but I was not embarrassed about it.

'Embarrassing' describes an event or situation that makes you feel very uncomfortable emotionally. Awkward can be a mild form of this but is also used to describe physical things; 'Carrying this table upstairs will be awkward.' An awkward person is someone who is a bit clumsy.