I'm a CJP. There's no different interface, and I don't get paid quicker, but I tend to have better access to communicating with advertisers, either through my CJP rep, or through the advertiser directly. Advertisers also run CJP promotions with increased commission rates. You also can get preferential treatment - One program I'm in just changed their policy to prohibit traffic from a certain channel due to the violations and issues they've had, but I was told that I was the only acception to the rule due to my CJP status.
Through the status, I've also gotten private offers for affiliate programs (usually CJ Media, shorter term programs) that are not available to the normal affiliate base, or permission to use an advertiser's display URL on search, whereas regular affiliates are forbidden from using it. I also got a free pass to a certain conference this year through that status.
I filled out the application once my commissions from CJ were over $10k for the third month in a row, but I had already gotten somewhat of a direct publisher rep when I hit about $8k. Part of the application process is also approval of your promotional methods, and they may look into your account history to see if you've had any issues with advertisers in the past.
