Ok, I would like to know the main differences between these tow models and if the Classic Custom comes with a 60's neck profile.

*If you know of any Les Paul under 2000 dlls with the 60's neck profile please let me know
This is all right on the Gibson site. Might want to spend a few minutes looking things up for yourself - we're not really here to google things for you. Highlights are the baked maple board and the 'upgrade' binding and headstock appointments on the CC. Yes, the CC has a 60s neck.

There are plenty of LPs at that price point with the 60s neck. Take a few minutes to check the Gibson site for specs, and look at prices on Google (the Gibson site ones are MSRP, not very accurate).
I checked Gibson's site before posting, but I couldn't find anything about the neck profiles (maybe i didn't checked well)
But, anyhow, thanks.
I'm gonna check the site again for more info.
Each model has a 'Specs' page. There's a button on the right side of the page with Home and Specs links, and sometimes another one or two. All the Specs pages list the neck profile and should help you find differences between the models.