Self-Publishing or Academic Publishing (on Springer) for Mathematics books?
The problem with self publishing with Amazon or otherwise is that no one will be able to find your book unless there are some special circumstances. I've done this as well as published with large publishers, including Springer.
If you are incredibly well known in your field and have a valuable web presence and anyone who might want to find your book can find you, then it can work and you can get a lot of sales and retain more of the revenue. But otherwise the book will just get lost in the millions of others. Publishers actually, take some effort to advertise and promote the books they publish, but it is much harder for you to do so, since people know about Springer, but maybe not about Joe Doakes and what he does.
For text-book material, the big companies will travel to book exhibits at conferences that educators might attend and give their (your) materials some visibility. They might also help you attend and meet some of your potential audience. But for professional works there is less opportunity for that.
The down side of the big publishers is that they can be abusive in some ways and I found that their promotions of books was very time limited. Two years. If you had a popular book they will want a new edition in two years. If it isn't popular, then it will just disappear.
But one valuable service of the big publishers is that they can help you a lot with copyediting and such. It is very hard, I find to edit one's own work. When you read, what you &quot;see&quot; is what you thought you wrote, not what you actually wrote. You can pay people for this, but they need to be math &quot;aware&quot; for it to work out.
A minor issue, perhaps, depending on your material, is that the big outfits are likely to use better materials, such as acid-free paper, than what the in-house Amazon system does. The same is true for the quality of bindings.

I'm working a bit on a math book myself. But it is a labor of love, when I can find the energy. But I expect exactly zero sales. Mathematicians are a weird lot, I guess.