What advantages and disadvantages should I consider in deciding whether to publish my academic book traditionally or self-publish?
Should I publish it traditionally or self-publish?


Why do you want to publish at all?  You answered


  I write the book to store down my research results and to spread my new knowledge. To make money is not the main aim, but it would be nice.


Given that: the answer is that you should certainly not self-publish your work.  You can store your results and spread knowledge by having the material freely available on the internet, as I believe is already the case.  The arxiv is one nice place to put work, but it is not the only one: you could put in on github or any number of other repositories.  You can just put it on your own website and make sure that google indexes it.  That means that billions of people can access it at any time.

Let me be clear with you: you are not going to make money self-publishing works of mathematics that you have not been able to publish traditionally.  It is exceedingly rare for any mathematical text beyond the undergraduate level to make a profit that is worth the time taken to write it.  (Maybe a few of Serge Lang's books qualify; probably not.)  If you go self-publishing rather than traditional publishing, you will lose money, and what you're paying for is the vanity of being a published author. 

The bar for interest by the mathematical community is much lower than the bar for the type of public interest needed to generate any real sales.  The thought that you have "My ideas are too bold for the mathematical community, so I need to take matters into my own hands; they don't know the value of my work as well as I do" is not only crankish but actually specifically damaging to you: it makes you ideal prey for predators of various kinds.  You told us in a previous question that you literally fell prey to a diploma mill and thereby lost money.  The same mindset that you have now is going to cost you more money in the future.  

I'm sorry to tell you this, but this has been going on for several years now, so I feel I should be plain: no one in the world has found your work to be of significant mathematical value.  This means that, with probability slightly less than one, that your work does not have significant mathematical value.  But in the unlikely event that your work does have value, you're not doing what is necessary in order to show it.  Mathematical research is not about simply writing down structures that generalize other structures and proving results about them.  You have to solve old problems or pose new ones that are of interest to the community. Bold statements of superiority would be a positive thing if they are specific and factual: for your work to be "superior", it should solve at least one problem that others have posed.  If you've done that, please explain yourself properly and then your work can be published in the mathematical mainstream.  If you haven't: please start to be honest with yourself about the value of your work.  Your livelihood is at stake.