cancel is intended for indicating mathematic cancellation, and consequently only works in math mode; it also strikes out diagnoally.
soul and ulem seem pretty similar. One basic semantic difference between them is that ulem per default changes the definition of \emph{...} to underline its argument instead of putting it in italics. This seems to aim at the reproduction of typewriter typesetting, when underlining often was the only means of indicating emphasis. Since using underlining for emphasis is mostly deprecated nowadays, I've always used soul when I did need to underline something.
The one big reason in favor of soul, however, is that it's able to deal with line breaks and hyphenation:
\documentclass{article}
\textwidth=1cm % just to force hyphenation to happen
\usepackage[T1]{fontenc}
\usepackage{cancel,soul,ulem}
\begin{document}
\[
\frac{\cancel{5}*17}{\cancel{5}*23}
\]
X \sout{supercalifragilisticexpialidocious}
X \st{supercalifragilisticexpialidocious}
X \emph{supercalifragilisticexpialidocious}
X \ul{supercalifragilisticexpialidocious}
\end{document}
(The X are necessary because LaTeX won't hyphenate the first word in a paragraph.)

ulemfeels like a tool for underlining etc. whereas theSOULpackage feels more like a general toolkit for creating text decoration macros that comes with a few pre-defined commands that show you how to use it. – Scott H. Oct 01 '12 at 16:42soulworks as far as I can see requires the overlay to be "horizontally uniform", so for wavy lines such as https://tex.stackexchange.com/questions/67064/a-thicker-wave-underline, you have to stick toulem(as mentioned below does not allow automatic hyphenation) orsoulpos(needs two compilation passes). – user202729 Sep 23 '22 at 08:01