Difference between “I will call you” and “I give you a call”?

Solution 1:

First of all, you can't say "I give you a call". I gather you meant "I will give you a call".

As far as phone calls are concerned, there is little, if any, difference in meaning or register. However, the first sentence:

I will call you.

is much more generic and, without proper context, may have meanings other than phone calls. For example, when you read

Go to your room. I'll call you when dinner is ready.

You don't imagine any phones involved, whereas

Go to your room. I'll give you a call when the dinner is ready

implies a strange situation when you make a phone call to someone in the same house.