Relation of Bayes Theorem, Conditional Probability and Likelihood

Bayes Theorem is derived using conditional probability:

$$P(A|B)\cdot P(B) = P(A) \cdot P(B|A)$$

Here terms are either probability or conditional probability. But later, in

$$P(A|B) = P(A)\cdot \frac{P(B|A)}{P(B)}$$

$$P(B|A)$$ is called likelihood.

Since likelihood and probability are not the same. Please help me understand: How did we end up with likelihood here and when do we use conditional probability and when do we use likelihood?

As you note, $$P(B|A)$$ plays two roles. The role depends on whether $$A$$ is fixed or $$B$$ is fixed.
When $$A$$ is fixed, $$P(B|A)$$ is the probability of $$B$$ given the value of $$A$$. Since probabilities add up to one, we have $$\sum_B P(B|A) = 1$$ for any value of $$A$$ for which the conditional probability is defined.
When $$B$$ is fixed, $$P(B|A)$$ is a function of $$A$$. This function is called the likelihood. Since likelihood is not probability, $$\sum_A P(B|A)$$ does not have to equal 1. It doesn't even have to be finite. Nevertheless, $$\sum_A P(B|A)\,P(A) = P(B)$$, which depends on $$B$$.
$$B$$ is fixed when Bayes theorem is used to compute the conditional probability $$P(A|B)$$: $$$$P(A|B) = \frac{P(B|A)\,P(A)}{P(B)} .$$$$ Therefore, $$P(B|A)$$ plays the likelihood role in this expression.
If instead we used Bayes theorem to compute the probability $$P(B|A)$$, the roles would be reversed and $$P(A|B)$$ would play the likelihood role: $$$$P(B|A) = \frac{P(A|B)\,P(B)}{P(A)} .$$$$