.jp domains can generally be obtained by anyone as long as one of the contacts has a Japan address. Often this is provided by proxy by the registrar.
.co.jp domains require you to have a registered company in Japan. The price is higher, and one company can only have one .co.jp domain.
When looking at the ranking of several KWs in Yahoo Japan the top spots tend to go to sites hosted in Japan regardless of whether they're .com, .jp or .co.jp. However, if you're going to host outside Japan, then it's advisable to get a local domain. I don't see many Japanese content sites hosted outside Japan on .com, .net or .org domains that place as well in the SERPs.
Years ago when there were more Japanese SEs some engines would determine locality primarily by a .jp domain, and then if that was lacking they'd simply look for Japanese content on the page. Hosting wasn't a part of the equation.
Google Japan does look at hosting locale when determining the target market of the site. However, these days you can use their webmaster tools to set that preference. You can't do that with Yahoo Japan though, and they're far and away the market leader.