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: 
I want to address this question by citing my current Master's advisor when I asked him the same question:

1. In the Bachelor's, you learn the basics of the area.
2. In the Master's, you learn what people are doing in the field and how to read scientific papers.
3. As a PhD, you learn how to develop your own research and start having new ideas on the field.
4. As a Post-Doc, you learn how to mentor other people's research while producing your own work without supervision.
5. When you reach Professorship you only have to look smart!
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.