the goo definition of 気恥ずかしい:
なんとなく恥ずかしい。きまりが悪い。
I don't really understand the difference.
「気」 is a prefix in 「気恥{きは}ずかしい」. It is read 「き」 or 「け」, depending on the word it is attached to.
This prefix adds the meaning of "somewhat", "kind of", "to an extent", etc. to the verb or adjective that it attaches to. Note that these are what 「なんとなく」 means at least roughly.
Thus, 「気恥ずかしい」 means "somewhat embarrassed", "a little bit ashamed", etc. whereas the plain 「恥ずかしい」 means "embarrassed", "ashamed", etc.
Common words using this prefix include:
「気{け}だるい」 = "somewhat dull", "kind of listless", etc.
「気忙{きぜわ}しい」 = "somewhat fidgety"
動詞・形容詞に付いて、なんとなく、漠然としたなどの意を表す。「気おされる」「気だるい」