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it is not peer reviewed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
A patent is a paper that describes an invention. When the patent is accepted, the invention is protected to the extend of the made claims. This means that you can sue people for replicating your invention into their product, within the scope of your patent.
The deal of patents is you get exclusivity in exchange of detailed public disclosure of the invention, which become free to be exploited after a certain period of time (namely 20 years?)
Research paper on the other hand, aim to disclose a finding regardless of any application. Its main focus is results of some experiment, which may or may not include products, inventions or applications of any type. The peer-review makes for more credibility to the content of the publication.
Making a publication after a patent should be possible, on the other end the opposite is not. While filling a patent, the invention must not have been already disclosed. Otherwise it is considered prior-art.
A patent is a legal document written by lawyers trying to say as little as possible while protecting an idea. Peer reviewed papers are trying to inform people and say as much as possible about an idea.
one makes you money, the other one keeps you a poor academic.