First, I warn you that this is my own personal interpretation.
At the time, we were preparing to send an unmanned probe to Venus (and this is mentioned in the speech). Alas, the Soviets did beat us to that one too (their program was ahead of ours in many aspects).
The speech is said to have defined the finish line for the Space Race (going to the Moon). However, if the "other thing" was also part of the finish line, then the Space Race (as defined in that speech) ended up in a tie.
We got the Moon, they got Venus.
However, it is not certain that "the other thing" was the launching of a probe to Venus.
In any case, when the speech is mentioned nowadays, the Moon part is enhanced (I say that this is because we won that part of the race) and the "other thing" is rarely explained (which makes me think that it is the Venus thing, which we did not win).