I don't really see the point of alternate fingerings when you could already use another.
When you learn to play the clarinet, or sax, or several other woodwinds, you'll soon find that it's much easier to play a variety of note sequences by making use of the alternate keys or alternate fingerings. In fact, if you look at the three articulated keys controlling the bottom holes (so either left or right pinky can play each of 3 notes), you'll quickly understand that many note sequences would be unplayable without this option.
I'm not sure which instrument you are asking about, however, the following applies to all instruments.
Sometimes the particular fingering for a particular note or set of noted depends on what comes before or what comes afterwards in the music. Music doesn't consist of single notes or single chords (except for some ultra-minimalist stuff maybe).
On a piano, one can play any isolated note with any finger. However, much of alternative fingering technique is designed to make the music sound smooth and to minimize the hand movement necessary.
On wind instruments, sometimes alternative fingerings have different tone quality and some may be more suitable for fast music and some for slow. Again, minimizing hand movement is still useful.
If you want to trill on a note, having to both put fingers down and up makes it really hard. So you often revert to an alternate fingering, sometimes with not really kosher pitch. Flutes and some other woodwinds actually have extra redundant keys specifically for facilitating trills.
Alternate fingerings not necessarily produce the identical tone, it may be slightly different, which depending on the context, may be reason enough to use it. So standard fingering may be slightly to sharp, which is fine for a leading tone, where this is exactly needed.
There also also some simple fingerings, which work fine for a single/starting tone, but require massive attention if applied to the end tone of a slur. So a fingering, which is slightly off the mark, but reliably works may be preferred.
For the mentioned trills are special trill tables, so I would not consider them as alternate fingerings in general, since due to the short duration of each note, bigger deviations from target pitch are acceptable.