I don't think that as asked the question is a good fit for the SE network (to be clear, I'm thinking about its fitness for TeX-SX, I don't think it would work in any way on MO). I think that the best strategy would be to set out to write a blog post about this and to ask specific questions when you find that you aren't able to figure something out for yourself. A lot is in the AMS guides, and more in standard typographical references.
Regarding what the actual differences are (there's some information as to why as well), here are some questions on TeX-SX where you could start reading.
- Use \langle, \rangle instead of <,> for inner products
Difference between \langle and <
- use \ll, \gg instead of <<, >>
How can I get double angle brackets, i.e. - << >>
- use \lvert, \rvert for absolute values, and \lVert, \rVert for norms
\mid, |, \vert, \lvert, \rvert
- use \mid instead of | for the "divides" symbol
\mid, |, \vert, \lvert, \rvert
- Use \operatorname for various operators such as Hom, End, etc.
What's the difference between \mathrm and \operatorname?
- Use \colon when declaring domain and range of functions, e.g. f \colon X \to Y
Using \colon or : in formulas?
- Use \dots instead of \cdots, \ldots for most ranges
How to type the three dots in a product?
- Use -- instead of - for page ranges and for joint authors (e.g. Cauchy--Schwarz)
Dashes: - vs. – vs. —
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