I would say there is a difference.
- I'm happy the way I am.
I'm happy the way I am means you aren't pressured to change something about yourself.
eg I'm not fat but I'm not super skinny, I'm happy the way I am. I'm not going to change anything about my appearance or how I look. Other people are not going to make me change myself.
- I'm pleased with myself
I managed to do something well, or achieve a goal I had, so I feel really good about myself.
eg I went to the market to buy all the food for dinner tonight, and I managed to get two steaks for $10! I'm really pleased with myself, I managed to bargain with the person (and get something that I really wanted, in this case cheaply)
I managed to get first place in the music competition, I'm really pleased with myself!