What genres do you listen to? If your music is a victim of the 'loudness war' digital is probably not the way to go(with cheap modern/computer digital it's either bright or mushy... bright + loudness = lots of fatigue.) Another factor is that records can either be significantly more expensive, significantly cheaper, or simply not available depending on your musical taste.
Personally I'd split your budget to ensure you can get the best recordings of your favorite albums(neither format is universally superior.) I'd say spend $200 and grab an old high end CD player off ebay, and put $400 into either a decent modern or good vintage table. Try to save at least $80 for the cartridge, based on your musical preference and my experience I'd say to go for a Nagaoka(or Grado if your musical tastes are what it excels with.)
Cheap computer audio will leave you disappointed imo. I do believe that with an unlimited budget the computer can overcome the disc spinner, but even my >$500 DAC, custom high performance computer(not built around audio though), and $150 Jplay software isn't as good as my old CD player in some regards. The CD player can be found for $50 on ebay too. Well, that is with bad recordings at least. Good records sound glorious on each format, though not always for the same reason. In my experience analog is king of soundstage and timbre, whilst digital is better for getting that 'you are there' feeling as ambient information comes through clearer.
And don't read too much into things, it is possible to tune either format to sound like the other. I'd have a hard time telling my old Dual701 with a cheap AT cartridge apart from the ASUS Essence STX I used to have. They were both very bright, and very dynamic sounding. Stereotypical 'digital' sound from the both of them. They were also about the same price.