Base Rate Fallacy:
This occurs when you estimate P(a|b) to be higher than it really is, because you didn’t take into account the low value (Base Rate) of P(a).
Example 1: Even if you are brilliant, you are not guaranteed to be admitted to Harvard: P(Admission|Brilliance) is low, because P(Admission) is low.
Example 2: The chance that you are an astronaut given that you live in the USA P(Astronaut|USA) is low, because the Base-Rate of Astronauts P(Astronaut) is very low.
Confusion of the Inverse Fallacy:
We often come to such false conclusions because we see that the inverse is true.
Most people who are admitted to Harvard are brilliant and most astronauts live in the USA.
However, this does not imply that if you are brilliant you will be admitted to Harvard or if you live in the USA you are an Astronaut.
When this occurs you are falsely assuming that P(a|b)=P(b|a) while P(a|b) and P(b|a) are unrelated.
Are they really different?
When the base-rates are equal P(a)=P(b) then P(a|b)=P(b|a). So if there are an equal amount of Americans and Astronauts then the following is valid: if most Astronauts are American then most Americans are Astronaut.
Therefore, another way to explain the Confusion of the Inverse Fallacy is by seeing that you falsely conclude that the base-rates are equal, while in reality one is bigger than the other. In other words: in both fallacies you don't take the Base Rate into account. Hence the confusion of the OP.
My conclusion is therefore that the Confusion of the Inverse Fallacy is a variant of the Base Rate Fallacy, which is more general.