Conditional probability - a formal discussion This is a rather philosophical question.

P[B|A] is formally defined as P[B and A]/P[B] where A and B are events in a sigma algebra and P is a probability mass function.

That is, P[B|A] is just a division of two numbers. If so, how come there are problems where we find it hard to calculate P[B and A] as well as P[B], but it is easy for us to reason about P[B|A] and so we assign a value to P[B|A] immediately without going through the division? (I can't think of an example for this, but I surely recall there are such cases. Can anyone share an example?)

To be more concrete, I'd be happy to see an example where it's hard\impossible to calculate P[A and B] or P[B] but it is easy to reason about P[A|B] on intuitive levels along with a justification for this reasoning (I'm talking about sample space and probability function definitions).
I think you mean $P(A|B)$ rather than $P(B|A)$; I'll assume that.

It might happen that event $B$, if it happens, controls the conditions for $A$ to happen, which does not imply that one has any idea of how probable $B$ is. As an extreme case, $B$ might logically imply $A$, in which case $P(A|B)=1$ regardless. Another example is if someone tosses a coin but I have no idea whether the coin is fair; for the events $A$: I win the toss, and $B$: the coin is fair, I know by definition that $P(A|B)=0.5$, even though I know nothing about $P(B)$ or $P(A\cap B)$.
Role of Conditional Probability - My thoughts:

A need for representing event in the presence of prior knowledge: Consider the probability of drawing a king of heart randomly from a standard deck of 52 cards. The probability of this event without any prior knowledge is 1/52. However, if one learns that the card drawn is red, then the probability of getting a king of heart becomes 1/26. Similarly, if one gathers knowledge that the card drawn is a face card (ace, king, queen or jack) then the probability gets shifted to 1/16. So we see that representing an event in the presence of prior knowledge is important and conditional event representation of (A|H) is the most adopted representation to solve this need. No matter what representation we adopt, we can agree that conditional event solves an unique concern of representing conditional knowledge.

What is most elemental - unconditional Vs. conditional: The debate whether conditional probability is more elemental than (unconditional) probability remains as an enticing subject for many statistician [1] as well as philosophers [2]. While the most adopted notation of conditional probability and its ratio representation viz. P(A|H)=P(AH)/P(H) where P(H)>0 indicates (unconditional) probability is more elemental; the other school of thoughts has their logic too. For, them, when we say probability of getting face value of 2 in a random throw of a fair dice is 1/6, we apply prior knowledge that all throw will lands perfectly on a face such that a face will be visible unambiguously, or that the dice will not break into pieces when rolled and so on. Therefore we apply a prior knowledge in order to determine a sample space of six face values. No matter what kind of probability is the most elemental, following the notation of conditional probability, we can agree that we speak of (unconditional) probability when we’ve accepted a sample space as the super most population and we’re not willing to get astray by adding further sample points to this space.  Similarly, we speak of conditional probability when we focus on an event with respect to the sub-population of the super-most (absolute in this sense) population. 

Is there any case which can be solved only by conditional probability: Once again, as long as we accept the ratio representation of the conditional probability, we see that conditional probability can be expressed in terms of unconditional probability. Thus, conceptually, any problem where conditional probability is used, can also be solved without use of conditional probability as well. However, we must appreciate that for cases where population and sub-population are not part of the same experiment, the use of conditional probability is really useful (not necessarily inevitable). To explain this further, in case of finding probability of a king of heart given that the card is red, we don’t really need conditional probability because the population of 52 cards and sub-population of 26 red cards are very clear to us. However, for cases such as applying a medicinal test on a cow to determine if it has mad-cow-disease, if we know false positive and false negative probabilities of the test, then to find out probability that a cow has disease given that it has tested positive, conditional probability can be used with great effect. If I may bring an analogy of ‘plus’ and ‘multiplication’ symbols of mathematics, we all know that any problem that uses multiplication symbol, can also be solved without it by mere use of ‘plus’ symbol. Similarly, in terms of solving problems, conditional probability can be avoided altogether just like multiplication symbol in mathematics. Still, we can appreciate the usefulness of conditional probability just like we can appreciate the use of multiplication in mathematics.

----------Bibliography------------
[1] H. Nguyen and C. Walker, “A history and introduction to the algebra of conditional events and probability logic,” Systems, Man and Cybernetics, IEEE Transactions on
   (Volume:24 , Issue: 12 ), pp. 1671 - 1675, Dec 1994.

[2] A. Hájek, “Conditional Probability,” Handbook of the Philosophy of Science. Volume 7: Philosophy of Statistics., vol. 7, p. 99, 2011.