When to use congruent vs approximately? What is the appropriate usage of the congruent  ($\cong$)  vs. the approximately ($\approx$) symbols?
 A: The notation $\cong$ corresponds to U+2245 APPROXIMATELY EQUAL TO “≅”, and $\approx$ corresponds to U+2248 ALMOST EQUAL TO “≈”. However, the Unicode names, written in all uppercase, are rather misleading here; and the names are really symbolic identifiers rather than normal names.
The meanings and uses of symbols are a matter of convention in mathematics, but as the reference notations, we can use those defined in the standard ISO 80000-2. According to it, U+2245 “≅” denotes the relations “is congruent to” and “is isomorphic to” for point sets (geometrical figures), and it is also used “for isomorphisms of mathematical structures”. So it has nothing to do with numerical approximations, for example.
The symbol U+2248 “≈” denotes, according to the standard, the relation “is approximately equal to”. This is an intentionally vague relation; the standard adds: “It depends on the user whether an approximation is sufficiently good. Equality is not excluded.”
A: From LaTeX point of view, both can be used whenever they are in math mode, either inline math mode or displayed math mode.
But from mathematics point of view, a $\cong$ implies equivalence of objects while a $\approx$ implies proximity of objects. More detailed explanation should be easy to find on the web.
