Difference between "≈", "≃", and "≅"

In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?

The Unicode standard lists all of them inside the Mathematical Operators Block.

• ≈ : ALMOST EQUAL TO (U+2248)
• ≃ : ASYMPTOTICALLY EQUAL TO (U+2243)
• ≅ : APPROXIMATELY EQUAL TO (U+2245)

The notations $\cong$ and $\simeq$ are not totally standardized. Both are usually used for "isomorphic" which means "the same in whatever context we are." For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in some senses they are "equivalent," though not always "equal:" you could have two congruent triangles at different places in a plane, so they wouldn't literally be "the same" but their intrinsic properties are the same. I've seen colleagues use both for isomorphic, and some (mostly the stable homotopy theorists I hang out with) will use $\cong$ for "homeomorphic" and $\simeq$ for "up to homotopy equivalence," but then others will use the same two symbols, for the same purposes, but reversing which gets which symbol.

The $\approx$ is used mostly in terms of numerical approximations, meaning that the values in questions are "close" to each other in whatever context one is working, and often it is less precise exactly how "close." Topologists also have a tendency to use $\approx$ for homeomorphic.

The main take-away from this answer: notation is not always standardized, and it's important to make sure you understand in whatever context you're working.

$\approx$ is used mostly for the approximate (read: calculated) value of a mathematical expression like $\pi \approx 3.14$ In LaTeX it is coded as \approx.

$\cong$ is used to show a congruency between two mathematical expressions, which could be geometrical, topological, and when using modulo arithmetic you can get different numbers that are congruent, e.g., $5 mod 3 \cong 11 mod 3$ (although this is also written as $\equiv$). In LaTeX it is coded as \cong.

$\sim$ is a similarity in geometry and can be used to show that two things are asymptotically equal (they become more equal as you increase a variable like $n$). This is a weaker statement than the other two. In LaTeX it is coded as \sim.

$\simeq$ is more of a grab-bag of meaning. In LaTeX it is coded as \simeq which means "similar equal" so it can be either, which might be appropriate in a certain situations.