How does the sound from neck pickup differ from the sound from bridge pickup in an electric guitar?
The existing answers all have part of the answer, however it is a bit more complex.
As No'am and teodozjan commented, the neck pickup sound is rounder and has more bass, and Tim mentioned vibrations at the neck being bigger so more bassy. But as you'll see from this picture (from Wikipedia) the pickups actually have a different balance of harmonics.
As an example, the middle pickup in this image has 2nd, 3rd and 4th harmonics at a higher amplitude than the fundamental, which changes the tonal quality significantly.
And the neck pickup has a very low 4th harmonic component.
So yes, the neck pickup is generally more bassy, but it's the balance of harmonics that really changes the sound. And that balance also changes as you move up the neck - less so for the bridge you can see from the picture, but very much for the neck (giving some wonderful tones with barre chords above 10th fret as used by many jazz players)
Pups pick up vibrations from the strings and turn them into electric signals going to the amp. If a pup is under a string where the vibration moves the string less, as in close to the end, it results in a loss of bass tone. So, even with identical pups, - as in the majority of guitars- the bridge pup is under the part of the string which vibrates less, for the want of a better word, than the neck pup, which is nearer the middle of the string, so will see larger vibrations. So, the sound from it will be mellower, more bassy, richer.
Add in the modern approach of using different pups in each place, often with phasing facility, and it's a whole new can of worms.
Sound of pickups changes in similar way to picking in different area on acoustic guitar.
Sound from bridge pickup has more treble and it is often used as lead tone and rhythm playing while playing heavily distorted.
Sound of neck pickup has more bass in it sound and is often used as clean rhythm and heavily distorted lead tone.
Gibson called neck pickup rhythm and bridge treble but this was misleading for beginners playing modern music.