TITLE: When should I use $=$ and $\equiv$?
QUESTION [27 upvotes]: In what context should I use $=$ and $\equiv$?
What is the precise difference?
Thanks!
(I wasn't sure what to tag this with, any suggestions?)

REPLY [19 votes]: The $\equiv$ symbol originally meant "is identically equal to", and as that name implies it is used with identities. It is actually stating that the equality holds for all instantiations of the free variables. For example $\sin\left(\theta+\frac{\pi}{2}\right)=\cos{\theta}$ is true for any value of $\theta$, therefore $\sin\left(\theta+\frac{\pi}{2}\right)\equiv\cos{\theta}$. 
People often got that confused with "equal by definition" or "defined to be". There are separate symbols for those meanings, including $\triangleq$ and ≝ (Unicode 0x225d). The $\equiv$ symbol has been used for this purpose so often that this is now sometimes considered a correct usage.
The $\equiv$ symbol was also repurposed to mean a congruence relationship like several of the other answers have discussed.