They seem exactly the same to me: when a variable is assigned a value, it’s equal to that value now.

Yeah it's confusing because in maths they are the same and use the same symbol but they are 100% not the same in programming, yet they confusingly used the same symbol. In fact they even used the mathematical equality symbol (=) for the thing that is least like equality (i.e. assignment).

To be fair not all languages made that mistake. There are a fair few where assignment is like

x := 20

Or

x <- 20

which is probably the most logical option because it really conveys the "store 20 in x" meaning.

Anyway on to your actual question... They definitely aren't the same in programming. Probably the simplest way to think of it is that assignment is a command: make these things equal! and equality is a question: are these things equal?

So for example equality will never mutate it's arguments. x == y will never change x or y because you're just asking "are they equal?". The value of that equality expression is a bool (true or false) so you can do something like:

a = (x == y)

x == y asks if they are equal and becomes a bool with the answer, and then the = stores that answer inside a.

In contrast = always mutates something. You can do this:

a = 3
a = 4
print(a)

And it will print 4. If you do this:

a = 3
a == 4
print(a)

It will (if the language doesn't complain at you for this mistake) print 3 because the == doesn't actually change a.