Beauty is one possible verdict; aesthetics is the whole apparatus of sensory judgement that produces verdicts. Baumgarten set beauty as his discipline's aim, but what he named was a science of sensory knowledge as such — and the field that grew out of it takes the ugly, the sublime and the merely vivid as seriously as the beautiful.
Side by side
Aspect
Aesthetic
Beauty
Scope
All sensory judgement
One verdict
Includes the ugly
Yes
No
Countable
Yes
Yes too — 'a beauty'
Named in
1735
Ancient
What each one means
Aesthetic
Relating to beauty, or pleasing in appearance — and, as a noun, either the philosophical discipline of sensory knowledge founded in 1735 or a particular conception of beauty. That last sense is long established (the Bauhaus aesthetic); the internet broadened it into today's countable noun for a coherent visual identity.