I am confused about the meaning of "use of internet" and "internet usage". Is there any difference between these two phrases? Does the phrase "use of internet" mean I utilize the internet or it means the benefit of using internet? And is it grammatically correct to use "usage of internet"?
The two phrases are both grammatically correct and equally acceptable. They mean the exact same thing: Both refer to utilizing the Internet and are neutral as to its benefits.
Sometimes internet usage will sound better than use of internet. Use of internet will give clarity in some contexts. Better listen to your lips. Use the phrase which sounds better.