~

Tilde (U+007E), primarily used as a diacritic (◌̃ [U+0303] or ˜ [U+02DC]), but sometimes used to denote omission of letters (i.e., a mark of suspension or contraction). The tilde doesn’t deserve its own key on the keyboard, but it has one because of the (obsolete) dead key mechanisms used for efficient typewriter design. As far as I’m concerned, the tilde has been decommissioned. One should never use this key unless using a typewriter. Or, I suppose, for Bash expansion of the home directory on UNIX systems, which (I must admit) is rather convenient.

∼

Sim (U+223C), short for “similar to,” commonly used in front of numbers to denote an imprecise uncertainty (e.g., “I have ∼3 months to live”). It is also often used to indicate that two things are equivalent in some way. It can be read as “behaves like” or “behaves according to,” or can be used to express approximations of size of a number; i.e., “the same order of magnitude.” In statistics, it implies something is asymptotically distributed, or one thing being distributed according to the other. Sim is also used to denote proportionality (not to be confused with ∝ [U+221D]).

Caveat: characters can differ in appearance vastly with different fonts. Using the semantically correct character might not always look the best.