Answer: When I was in school we sang with numbers, Solfege being out of style with my teachers. But numbers are harder to sing, and unless you give up the logic of altered tones altogether they also have no graceful way to express alteration.

Say you have a piece in the key of D major. The notes natural to the key will be:

D, E, F#, G, A, B, C#, D

Assuming the "moveable Do" system in which the tonic (1) is always Do, those can be called

Do, Re, Mi, Fa, Sol, La, Ti, Do

or

1, 2, 3, 4, 5, 6, 7, 1

But just trying singing something like "5, 7, 6, 7..." and then sing "Sol, Ti, La, Ti" for comparison.

There's more: Solfege has special forms of these syllables that can be used if the note is sharped or flatted from what it would normally be in the key. For instance, the flat form of Mi is Me (pronounced "May"), and the sharped form of Re is Ri. With numbers you'd have to sing "3 flat" and "2 sharp", which is even clumsier than the uninflected numbers. You could give up the logic of diatonic music altogether and try using 12 numbers for the octave - but then you're stuck with singing things like "eleven." Plus which, leaving behind the tonal logic of western music would be kind of a shame - Solfege can tell you that a certain note is meant to be a raised sixth rather than a flatted seventh, and that tells you what that note is "all about." Most of the music people actually sing is tonal or modal, and the logic of such music shines through in Solfege.

As for where the syllables come from, there's another answer that has some background: check Question 65.