You likely only repeat if you fail
Narratively if the target fails the first save, it will be Incapacitated because it is drowsy and cannot act normally any more, and then, if it fails the second save, if will fall fully asleep and be incapacitated.
Comparison with other spells: This also seems similar to the flesh to stone spell, where a target first has to make one save to avoid being Restrained, and then, if it failed that save, another series of saves to not become Petrified. There, if you make the first save, there is no risk of becoming petrified, so it seems this should work in a similar way. However, the language of flesh to stone is different, it says
A Restrained target makes another Constitution saving throw [...]
Description Text: Other than in flesh to stone, the sleep spell does not say that an Incapacitated target makes another saving throw, it only says that the target will have that condition if it fails the first save:
Each creature of your choice in a 5-foot-radius Sphere centered on a point within range must succeed on a Wisdom saving throw or have the Incapacitated condition until the end of its next turn
It then say that at that point, the target makes another save:
at which point it must repeat the save
There is no "the Incapacitated target then...". The question is how to read at that point.
If you read it as "at that point in time", the spell merely talks about the time at which the second save has to be made, in which case the two saves are entirely unrelated, and even if you made the first save, and still could act normally, you will have to make a second save at the end of your next turn, or become Incapacitated (i.e. fall asleep). You get put to sleep if you fail that second save, no matter what.
If you read it as "at that point of what's happening here", you are incapacitated because you failed the save, and only if that happened, now that you are reaching the end of your next turn, you have to make a second save.
It might be possible that a DM makes a different call here, but considering the narrative logic, considering it works in a similar way for other spells that impose conditions of increasing severity, and considering that all of this is in one single sentence, I believe it is much more likely that the intent is that you only make a second save if you failed the first one.