There have been many threads about esquires. Do a search and have fun reading.
In my opinion the advantages of an esquire are:
1) the lack of a pickup, and therefore the lack of a magnetic field, near the neck where the strings are vibrating most freely (especially when you are playing a solo). If the strings aren't being 'dampened' by magnets then you get a bit more sustain and more harmonics, ie, a better more 'musical' sound. The difference is subtle, but playing hard thru a turned up tube amp it makes a difference, I find.
2) Having the 3-way free to allow a tone pot bypass (or even a tone and volume pot bypass straight to the output) adds another subtle, but helpful, edge to the sound.
3) the simplicity of only having one pickup
forces you to rely on your playing to deal with different musical settings. Some players like the challenge.
Disadvantages are:
1) you don't have the gorgeous telecaster neck pickup and 'both on' sounds. If you like 'em and use 'em you'll miss 'em.
Personally I find that in some venues I
can use the bridge pickup all night, but I don't really
like to. And in others places the acoustics are such that I don't much use the bridge pickup and really
need the other settings and being
stuck with an esquire all night would be hell.
So although I can hear and appreciate the 'esquire difference', in practice I just find it too limiting. The bands I currently play in aren't suitable for a bridge pickup only guitar, I think.
So I've added a neck pickup to mine and made it a nice telecaster.
(The Classic 50s Esquire I have is generally too good a guitar to just sit in the case and rarely be used.)