What is the mathematical difference between group and category? This question is quite similar to the following link:
Why learn Category Theory in order to study Group Theory?
The above link is nice but I could not find the difference mathematically between category and group. I can understand that answer to my question lies on the link but few comments on the difference perspective may help me.
 A: A group is essentially the same thing as a category satisfying the following two conditions: 1) there is only one object; and 2) every morphism is an isomorphism. Of course not every category has these two properties, so there are many more categories than groups (e.g., the category of sets, the category of categories, the category of groups, etc.). 
The precise meaning of the essential sameness above is that there is an equivalence of categories between the category of groups and homomorphisms (defined by the usual set of axioms) and the category of all single-object categories in which every morphism is an isomorphism and functors. There is no isomorphism between the two categories, only an equivalence. 
Informally, you can think of going from groups to all categories by first allowing any collection of objects but still requiring all morphisms to be isomorphisms (such a structure is called a groupoid), and then allowing non-iso morphisms (to get a category). A group is the mathematical representation of symmetry of an object. A groupoid is the mathemaical representation of symmetries of objects, and the way the symmetric interact. A category then is no longer representing symmetry, but rather simply representing structure preservation in the broadest sense. 
There are obvious inclusions of groups in groupoids in categories, and you may wish to contemplate left and/or right adjoints of these to appreciate the above remarks. 
