I feel awkward.... I feel self-conscious...
They don't mean exactly the same thing but there is a lot of overlap..
I feel awkward with him suggests there is possibly something about him that is making me uncomfortable. He makes me uneasy, creeps me out, gives me the willies. Or perhaps something unpleasant happened in the past between the two of us that makes it an awkward situation. It could have been either something I had done or he had done, something embarrassing or shameful for example. Now it's awkward when ever I am around him.
If I feel self conscious it probably has something to do with me, not him. For example, I might be thinking about my appearance or concerned that I am coming across as too nervous or anxious. Here, I am concerned specifically with how I am being perceived by him or others.
"I always feel awkward with I am with him." = If I said that I would feel uncomfortable with my body or my words. I would feel that my words and body just aren't working smoothly or what I do is a little bit wrong. I am probably doing something wrong and I feel uncomfortable.
"I always feel self-conscious when I'm with him." = If I said that I would feel that I was thinking about my self too much. This could be my body or my words. But it could also be my clothes, my food, my beliefs, my anything. I might not even be doing anything wrong, but I am thinking very much about all parts of my self and I am not comfortable.
But they are basically the same thing. "I always feel uncomfortable when I am with him."
However, now that I wrote that...
"I always feel uncomfortable when I am with him." = If I said that it could come from many different sources. Maybe I feel awkward. Maybe I feel self-conscious. But it could also be him and not me. He could be rude, dangerous, awkward, insecure, a different social status, loud, talk too much or not enough, never offers a drink or food when you are a guest. There a lot of different things can make a person feel uncomfortable.
I should add that I am making all this up as a native speaker. I am not a linguist. I am American and I enjoy English.
They can mean the same thing, but not always. Awkward has a less specific meaning. Feeling awkward basically means feeling strange or uncomfortable. Self-conscious is a bit more specific. Feeling self-conscious means that you feel insecure or you are worried about how you will be perceived by other people.