Respected sir, what is the difference …

Proper Subset is a subset of the set which is having the elemnts from the Main set but not all. It is 2ⁿ - 1 in numbers.

Non Empty Proper subsets are all the subsets excluding the "Empty Subset" i.e. { }. It is 2ⁿ - 2 in numbers.

For e.g. If A = {1,2} then the number of subsets it has is 2ⁿ where n represents the no. of elements in that set means A will have 2² = 4 subsets which are {1}, {2}, {1,2} & { } sets. 

Now, here {1}, {2} & { } are the proper subsets & {1}, {2} are the non empty proper subsets.

Hope it clears better.

If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B. 

E.g

A = {1, 2, 3, 4}

Here n(A) = 4

B = {1, 2, 3, 4, 5} 

Here n(B) = 5 

We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A. 

So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B