If A & B are independent events and the order in which they happen doesn't matter, then why is P(B|A) is not the same as P(A∩B)?
P(A∩B) denotes the probability of A & B happening together.
P(B|A) denotes the probability that B would happen given that A has already happened. In essence, both A & B must happen and since both are independent events, P(B|A) should equal the probability of intersection.
What am I getting wrong here?
If $A$ and $B$ are independent events, then you expect that the occurrence of $B$ is not affected by the occurrence of $A$, and vice-versa. For example, suppose we roll two dice. The first number is not affected by the second one, nor the second one is affected by the first one. This is independence. This is why we write, for independent events: $$P(B|A) = P(B) = \frac{P(A\cap B)}{P(A)},$$ because $A$ does not influence $B$ to happen, so $P(B|A)=P(B)$. Now, $A\cap B$ is interpreted as the occurrence of $\textbf{both}$ events $A$ and $B$, and this is rather different than independence.