-면서 vs -ㄴ/은/는 김에
The two are really different expressions with very little in common.
* -면서 - "while ...". Expresses doing/being one thing while also doing/being another at the same time.
* -ㄴ/은/는 김에 - "Since/As I am doing something, (I might as well)" or "While we're doing something ...". Expresses doing something unplanned because another action/situation made it opportune.
Examples:
1 저녁을 먹으면서 신문을 본다 - Read newspaper while eating dinner.
2 그는 일등을 했다면서 자랑한다 - He brags saying he won the first place in the contest.
3 밖에 나온 김에 아예 장도 보자 - Since/While we're out here already, might as well get the grocery too.
4 복강 절개를 한 김에 맹장 수술까지 했다 - Since/While my abdomen was opened up anyway (for some other operation), I also had an appendectomy done.