What's the difference between 'group' and 'grouping'?

When you use the word group your emphasis should be on the collection of things in the group.

How many people do you have in your group?

When you use the word grouping your emphasis should be on the act of forming the group rather than the group itself.

Which grouping would be better- girls in one group and guys in another, or else adults vs children?

Perhaps the thing that is confusing, is that "group" can be both a noun and a verb, and "grouping" likewise.

"Group" as a noun means the collection of things.
"Grouping" as a noun, is a gerund. A Gerund, in English is :"As applied to English, it refers to the usage of a verb (in its -ing form) as a noun." So, it's a verb being used as a noun! Maybe that's what's confusing you?

There is no difference in meaning whatsoever between these two nouns. The only difference, is their usage. That is, there are situations where one would use "group", but not "grouping" E.g.:

The group of people gradually grew in size.
I don't like the grouping system we have at work....

We cannot say, "I don't like the group system..." It's grammatically incorrect.

"Group" as a verb is to arrange into a group. To form a collection of people or items.
"Grouping" as a verb is the act of arranging the group in present time, E.g.: "I am grouping these letters together." Instead of "I am group..."

I hope that clears up any ambiguity.