What’s the difference between tilde(~) and caret(^) in package.json?

Question

See the NPM docs and semver docs:

~version “Approximately equivalent to version”, will update you to all future patch versions, without incrementing the minor version. ~1.2.3 will use releases from 1.2.3 to <1.3.0.
^version “Compatible with version”, will update you to all future minor/patch versions, without incrementing the major version. ^2.3.4 will use releases from 2.3.4 to <3.0.0.

See Comments below for exceptions, in particular for pre-one versions, such as ^0.2.3