"Lucky" means (something good) has happened or was given to you as the result of random chance. Often "blessed" connotes that something good happened or was given to you purposefully, like a gift from God, or from some other loving source. "Luck" sounds mechanical and without feeling; "blessed" sounds warm and wonderful.

I think the only difference is in style. "I'm lucky" sounds neutral, it's common for language we use every day. "I'm blessed" is an elevated phrase and is typical of language found in literature. Using such a phrase you emphasize that it is also very important to you.