Seem that the commandment is to count, why do we make a distinction between counting during the day and at night?
The Torah commands us to count the Omer "מהחל חרמש בקמה" (Devarim 16:9), ("When the sickle 'begins' with the stalks"). Thus, it is assumed that the Omer should be counted when the stalks for the Korban Omer should be cropped.
The mitzva of קצירת העומר (cropping of the omer) applies at night (Menachos 71a), and it is disputed amongst the Rishonim whether it is acceptable if done by day.
Thus, the Shulchan Aruch (OC 489:7-8) rules (following the Rosh) that if one forgot at night, he should count by day, but without a b'racha (see Tur OC, 489).
I heard a shiur last week where it was related to the word תמימות/complete (in שבע שבתות תמימות תהיינה/they should be seven complete weeks). A halakhic day begins at night, so in order for it to be counted "completely," the counting needs to be done at night.
According to R' Soloveitchik, when you "make up" a missed night by counting during the day, you aren't actually fulfilling the mitzva; rather, you are just enabling yourself to count with a bracha on subsequent nights. "Counting" can only be done if you count consecutively, so if you count 1 and skip 2, you can't then count 3. By counting 2 during the day, you've maintained the continuity so that you can still count 3.
While not a mainstream practice, there is a Tosfos on Menachos 66a that states that a person is allowed to make a bracha & count sefirah when there's still light outside (ie when it's doubtful whether it is nighttime yet).
Tosfos (ibid) s.v. "zecher l'mikdash hu" begins his commentary by stating:
נראה דבספק חשיכה יכול לברך ואין צריך להמתין עד שיהא ודאי לילה כיון שהוא ספיקא דרבנן
It seems that when it is doubtful whether it's dark one can make a bracha [for Sefiras Ha'omer], and a person does not need to wait until he's sure it is definitely night, since it is a safek mi'drabanan