They are very similar and the meanings overlap.
When you are embarrassed, you just feel self-conscious. You feel you have done something awkward, You don't think you've done anything wrong, or not seriously wrong. For example, "Because it was an important meeting, I wore a coat and tie (a full business suit). When I got there, I found that everybody else was wearing a tee-shirt and blue jeans. I was the only person wearing a tie. I was embarrassed."
Or, "I was trying to impress a date by taking her to a good restaurant. We drank champagne. Suddenly, I belched. That was embarrassing!"
When you are ashamed, you feel self-conscious _because_ you have been caught in public doing something that is actually _wrong._ You've done something wrong and everyone knows it.
"Annie, why is Johnny crying again? You took his toy away from him, didn't you!"
"I'm sorry, Mama."
Annie feels ashamed.