Question: I am a PhD student in Mathematics. I wanted to know what exactly is the difference between PhD and Postdoc. Is it just the research that one does after PhD(kind of second PhD?)?

Answer: 
The first and foremost difference between the two is that a PhD is "awarded" after defending a thesis (plus additional duties depending on the department). On the other hand, a PostDoc is a temporary working position that is assigned by some institution, whose completion does not require any defence.

What people do or do not do, researchwise, in their PhD and PostDocs is very much dependent on the area and on the department (you may look at other answers that go more in details on this and that I do not want to overwrite, as I would just copy and paste them).
This is akin to asking what's the difference between research master's and PhD, or Assistant to associate professor.

A postdoc, is the *next level* after PhD (though not a necessary one). This is the main point.

Practically, it is different, in that it is less formal, with less specific requirements, and you are paid for it, as you are employed, not a student, as other answers here explained well.