What are the differences between a research paper and a patent?
You're asking multiple questions at once, so let's start with:


  What are the differences between a research paper and a patent


There are many, but briefly a research paper has the purpose of communicating research findings to the relevant scientific community and the general public. Patents are legal documents used to prevent other people from commercializing the process or devices they describe. Research papers typically report results of a scientific process, whether experimental or not, while patents describe processes and devices along with their respective intended applications.

Patents are typically very open-ended, not unlike a very long "future works"-section of an academic paper. The strategy is to cover a maximum of possible usages for the technology.


  it is not peer reviewed


That is correct, peer review is a strategy used by researchers to filter, improve, and curate scientific literature. For patents, the review is driven by the legal requirements only. A major point is absence of overlap with existing patents and other publicly available material*. In some regions, a minimal demonstration of how the process/device works is expected by patent offices, but it's not the same as peer review.


  Is it allowed to publish patent and then paper? 


Yes, it's the other way that's problematic because as stated above, if you (or someone else) published a paper about your process or device you cannot patent it anymore.

Most journals are interested in novel results and all (legitimate) journals don't want to re-publish existing literature but the fact that the processes are patented is not an issue. As stated in my first paragraph, both documents should have very little overlap.



*On that subject, see the entertaining if not entirely historically accurate 
Donald Duck as prior art case