He embraced her, kissing her on both cheeks.
Question: I found this example sentence above from a dictionary. And I’m wondering why the word “kiss” with -ing form, and in front of kissing with a comma?
Is it okay to say, "He embraced her and kissed her on both cheeks? Does it make any difference?
Thank you!!!
There can be a slight difference since tenses are involved. So ‘He embraced her, kissing her on both cheeks’ would mean that during the embrace he was kissing her. As opposed to ‘He embraced her and kissed her on both cheeks’ could mean that after he embraced her he kissed her. They’re practically the same so you can use either! Hope this makes sense 
Actually it does.
“And kissed her on both cheeks” means that he rather kissed her one time on each cheek(that is twice in sum). It would look like a formality.
“Kissing her on both cheeks” means an undefined amount of times, because it doesn’t matter, so thrilled he was. Process fits better for expressing feelings.
This link might be helpful if you want a bit of background on the grammar:
Scroll down ultimately to the “describing two actions at the same time”
Also:
Scroll to “not acting like a noun, it’s a particle phrase section”.
So in your example it’s saying the action of kissing her on the cheek is happening at the same time as he embraces her, but it’s also acting as an “adverb” further describing the action of his embrace…i.e. “how” he embraced…kissing her on both cheeks.
Your alternative could be describing similar simultaneous action, but it could also imply that the man embraced her first and then kissed her on both cheeks. The original version doesn’t leave this ambiguity.
Ultimately they are saying similar things.
Thanks so much for the helpful links.
One more question. Is it okay to omit the punctuation, comma, in this sentence?
I don’t believe you can leave it out (original sentence).
The other thing I’d offer is the “embraced and kissed” feels a bit robotic where “embraced, kissing” feels a bit warmer and literary.