Unable to understand the difference between the two questions.

Let $T$ be the event that there are two boys, $O$ be the event that we observed a boy and $A$ be the event that there is at least one boy.

Then we have $$P(T|O)=\frac{P(O|T)P(T)}{P(O)}$$

$$P(T|A)=\frac{P(A|T)P(T)}{P(A)}$$

It is easy to see that the numerators in those statements are the same, i.e. $(1)(\frac{1}{4})=\frac{1}{4}$. The difference lies in the denominators.

The probability that we observe a boy is $$P(O)=P(O|BB)P(BB) + P(O|BG)P(BG) + P(O|GB)P(GB) + P(O|GG)P(GG)$$

The probability that there is at least one boy is $$P(A)=P(A|BB)P(BB) + P(A|BG)P(BG) + P(A|GB)P(GB) + P(A|GG)P(GG)$$

Notice the events where there is one boy and one girl. The probability that we observe a boy in those cases is only $\frac{1}{2}$ but the probability that there is at least one boy in those cases is $1$.

Therefore, $P(A)>P(O)$ so $P(T|O)\gt P(T|A)$.