Tilde "~" I use to state a geometric shape is similar to another one ie a triangle of sides 3/4/5 is similar to a triangle with sides 30/40/50. The approximation sign "≈" I use for decimal approximations with tilde "~" being a rougher approximation. In my work "=" is the identity of a number so it states an equivalence.
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. 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.
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. 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. The notations $\cong$ and $\simeq$ are not totally standardized.
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