I have a bunch of basses in my stable, and I think the biggest difference between an Alembic and some other marques are:
1. Bright, clear tone.
2. Versatility.
3. Adjustability (for playing ease).
4. Quietness.
5. Build Quality (second to none in my experience).
Can I pick out an Alembic from most basses? Most of the time, yes. Can I also pick out a P-Bass or a bass like a P-Bass? Yep.
As far as whether to use an Alembic or not, well, it depends on the gig. I've used Alembics for country, rock, blues, etc. and it's always fit the bill. I couldn't use, say, my Rick 4001s for blues - for me (and most likely the band) it just wouldn't work. However, I've used my Fenders, Lakland, Pedulla, etc. for other gigs and they work great as well.
I play Alembic because of this - they are simply the finest bass guitars on the planet, for me anyway. I love my Lakland, my '73 Jazz, my Sadowsky Vintage 4, my Pedulla MVP, but there's just something about an Alembic that just moves me.
Can others tell the difference when I'm playing an Alembic? Absolutely. I've had more than one front of the house engineer tell me that he'd never encountered such a pure, quiet, and clean bass tone. One told me it was the easiest bass to mix he'd ever encountered in 20 years as a sound guy.
Alan