"If I make up my own work, but base them on another work, my new work belongs to me."
False. U.S. Copyright law is quite explicit that the making of what are called 'derivative works' -- works based or derived from another copyrighted work -- is the exclusive province of the owner of the original work. This is true even though the making of these new works is a highly creative process. If you create a work using settings or elements from somebody else's work, you need that author's permission.
It's still very shady as how far that law will go - shady enough that most legal matters will stear well clear of 'derived theivery' cases if they get the chance.
For example. Artist A sees that Artist B has painted a painting using an airbrush onto glass. Artist A decides that this is a good idea and copies the idea from Artist B.
Under that explination, Artist A would be producing a piece of derivative work, and therefore would require the authors permission to do it. There is no clear line at what point it is acceptable to borrow ideas. Under perfect law, you wouldn't be able to take a photo of a tree from a 30 degree angle if somebody else did it, as they would legally be able to claim that you used their idea.
Theres also the issue of prooving that it was stolen. With so many songs out there, theres a suprisingly high chance of the same melody being accidently repeated into another song. That is the only real explination I can offer for this instance. To proove somebody stole something from your song, it dosn't just have to sound similar, you have to proove that 1: you made it and 2: that it really is stolen, and not just similar.
Again this is what really protects people who do covers and re-recordings. Think about how many bands cover big-popular bands tracks. Although it might be attractive for the recording companies to farm in money off of it, the costs of simply hunting down people who perform music under their lable is too expensive to be worth it.
To be honest, the copyright laws are only really a worrying issue if you are either making money off of somebody elses work, composition or not, or you somehow get amazingly popular - at which point its more likely the record company will try to buy the track off you or win it in a lawsuit so they can use it for their own means. There are hundreds of covers on NG, and this similarity has only surfaced because it was a popular artist. Still, I'd like to hear what Dim himself has to say about this.
Also, regarding video game covers or remixes. Although under the copyright law, they're basically illegal, but most of the copyrights on the tracks have expired regardless. Also Nintendo has shown its support for fansites, fan fiction and fan music - under similar copyright laws, even fan websites are illegal if you are using copyrighted names without permission!
Newgrounds has a remarkably loose rule system to protect from copyrights - normally only tracks which contain material in sound form from another track will be even considered for deletion. Newgrounds also does not have legal responsibility for the individual track so to be honest they don't honestly need to do that - Its most likely just done to prevent other users doing the same thing.
Also, performing an action like banning (heaven forbid) dimrain for submission of a track which is questionably 'stolen' (though 99% of it sounds nothing like the origional, it's not even the same genre for fucks sake) would also result in a huge backlog of other tracks that, for fairness, would also need to be removed. Mostly covers and remixes.