Should I use ~ or \mbox?

In none of the presented cases you should use \mbox.

• Phone numbers should use a kern, such as 123\,456\,7890

• Things such as p.~210~sq.\@ should use a tie (note the \@ in order not to make the period as a sentence ending one)

• Names should use ~: Jean de~La~Fontaine (you may want to remove the first ~ if line breaking becomes otherwise unfeasible)

• Space before French punctuations (but after «) should be a kern; the only exception is the colon, where ~:<space> should be used.

For French punctuation, leave the job to babel-french that usually works well.

What's the difference between p.~210 and \mbox{p. 210}? In the former case, the space is not considered to be after a sentence ending period and it still participates to stretching and shrinking together with all other spaces in the line. In the latter case, the space is fixed and . is sentence ending; so it will probably be much bigger than other spaces in the line. You might cure this by doing \mbox{p.\@ 210} or \mbox{.\ 210}, but still this space will usually be different from other spaces in the same line, because it's rigid.

The \mbox is totally unbreakable, i.e., it does not allow hyphenation nor any other breaks.

The tilde ~ inserts an unbreakable space, but does not affect the breakability of its left and right neighbours. So in La~Fontaine the Fountaine part can still be hyphenated.

So, the tilde is easier to type and to read in the TeX source and it has the other advantage that its neighbours can be hyphenated.