As far as I understand, The difference is "habit" not semantic. They mean exactly the same. But 什么也 are usually used in negative while 什么都 can be in both positive and negative.
You can't say 他什么菜也吃 is wrong, just not generally used.
My advice is simple: stick to 什么都 because it can be used both positive and negative, but understand that some people may prefer 什么也 in negative.
Actually, this is generally the principle when facing many options in expressions, for example, the 7 fundamental ways of continuous forms be + ~ing, (在+..., 正+..., ......呢) choose one of them that you feel most comfortably and most practical in using (in this case, my recommendation is 正在 because you can include place terms easily), and understand there are other 6 ways.