The difference between “mum” and “mom” is the same as the one between “colour” and “color”—the former is used predominantly in British English (and in other Commonwealth nations) and the latter in American English.
However, just like with “colour”, there is not only a difference in spelling but also a difference in pronunciation. Americans pronounce “mom” as /mɑːm/ (with the same vowel as in “father”), while the British pronounce “mum” as /mʌm/ (with the same vowel as in the word “but”).
There are also a handful of British dialects in which the word “mom” (not “mum”) is used, pronounced /mɒm/ (with a short “o” sound). It is worth noting that “mom” has started to appear more regularly in British literature, at least if the Google Ngram Viewer for British English is to be believed (the following diagram shows the relative frequency of the words “mum” and “mom” in British literature):