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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?)?

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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!
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    No, when you reach full professorship you only have to look smart. There's still "assistant professor" and "associate professor" to get through.
    – JeffE
    Commented Mar 4, 2018 at 0:19
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    True, this was a very simplistic point of view but useful one I think...
    – Jsf73
    Commented Mar 4, 2018 at 14:13
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In mathematics, a "postdoc" is a special kind of faculty job you get just after your Ph.D. It may have a fixed term (not tenure track). It may have reduced teaching, so that you can concentrate on your research. You may have a "mentor" assigned to advise you on doing research.

But (unlike some other fields like experimental physics) you are probably not working in someone else's "lab" on someone else's research program that someone else got the funding for.

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I have always understood a postdoc to be a "post doctoral" position, that is a paid post. The idea of a postdoc position is to do research roughly in the area of the PhD, with the person gaining experience that would possibly be of help in gaining a full time position.

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    Around here, PhD students also earn wages (unless they support themselves with a grant).
    – Tommi
    Commented Mar 5, 2018 at 13:09
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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).

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A postdoc is expected to know what they are doing, and to be productive researchers with minimal supervision. PhD students are learning how to be researchers: by the end of the process, they should be on par with a postdoc, but it will take time to get there

Edit: This comment has been copied from Barbara Robson's answer here.

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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.

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