The difference is in formality. 함께 is more formal that 같이.

You're gonna hear this answer a lot as you learn vocabulary. They're the same!

같이 is also used more in conversation while the other is more common in writing

같이 is used more in conversatuons, while the other is more common in writing

It can. "함께 먹읍시다." is also accepted.

함께/같이 먹읍시다, 먹자, 먹어요 / 드시죠 / 식사합시다, 식사하자, 식사해요 These all form can be used.

I have the same question

I wanted to ask that too. As 식사 is 食事, I'd say it's more like dining or having a meal.