Subsets vs Proper Subsets
It is quite natural to realize the world through categorization of things into groups. This is the basis of mathematical concept called ‘Set Theory’. The set theory was developed in the late nineteenth century, and now, it is omnipresent in mathematics. Nearly all of mathematics can be derived using set theory as the foundation. The application of set theory ranges from abstract mathematics to all subjects in the tangible physical world.
Subset and Proper Subset are two terminologies often used in the Set Theory to introduce relationships between sets.
If each element in a set A is also a member of a set B, then set A is called a subset of B. This also can be read as “A is contained in B”. More formally, A is a subset of B, denoted by A⊆B if, x∈A implies x∈B.
Any set itself is a sub set of the same set, because, obviously, any element that is in a set will also be in the same set. We say “A is a proper subset of B” if, A is a subset of B but, A is not equal to B. To denote that A is a proper sub set of B we use the notation A⊂B. For example, the set {1,2} has 4 subsets, but only 3 proper subsets. Because {1,2} is a subset but not a proper subset of {1,2}.
If a set is a proper subset of another set, it is always a subset of that set, (i.e. if A is a proper subset of B, it implies that A is a subset of B). But there can be subsets, which are not proper subsets of their superset. If two sets are equal, then they are subsets of one another, but not proper subset of one another.
In brief:  If A is a subset of B then A and B can be equal.  If A is a proper subset of B then A cannot be equal to B.
