When to use the rule of addition (Statistics)

There are two rules of addition in statistics:
P(A or B) = P(A) + P(B)
and
P(A or B) = P(A) + P(B) - P(A and B)

How do I know when one over the other?

they're both basically the same thing however you use P(A or B) = P(A) + P(B) when the two things you're comparing are Independent.
Independent simply means that the outcome of one doesn't effect the outcome of the other.
P(A or B) = P(A) + P(B) - P(A and B) is used when the objects aren't independent...
A trick to see if your items are independent is
P(A) * P(B) = P(A and B), if this is true then it's independent

If A and B are mutually exclusive events:

P(A⋃B) = P(A) + P(B)

If A and B aren't mutually exclusive events:

P(A⋃B) = P(A) + P(B) - P(A⋂B)