What is the difference between equal ($=$) and colons equal($:=$)?

Question

From the context I guess colons equals add the implication of setting. As in $t:=0$ means lets set $t=0$.

Answer 1

The $t:=0$ notation is borrowed from some computer programming languages. It implies we define $t$ to be $0$, so it's not an equation to be solved e.g. or part of some axiom, but an "assignment" of values.

Answer 2

The latter is used to define something. For example we have that $2 + 4 = 6$. That is just a normal equality. But if we have something that we want to define then we can use "$:=$". For example we set $A := \lbrace \text{apple},\text{banana} \rbrace$.