I've had both and wear both currently.
New Rocks are heavy, very gothy and DO create some stigma - people WILL assume you're a goth if you so much as look at a pair. If that doesn't bother you then it's all good.
New Rocks are NOT good quality at all and all the pairs I've had 4/5 over the years, have all come apart after around 2 years. Metal bits (that aren't actually metal, but resin) fall off etc.
Again, if that fact doesn't bother you, go for it!
DM's however, are one of the best investments you'll ever buy. They will last you at least 10 years, some people have them for 30+. There's a whole lot more choice when it comes to DM's - I've just bought my 27 year old sister a pair of pink flocked DM Mary Janes for her birthday. She's what I call a 'chav' so that says something about DMs and their versatility.
I love DM's above NR's. I used to love my NR's more, but that was when I was more gothy (never been the black lipstick velvet and lace type goth though) so now my DM's are a good inbetweener and look far less 'cliche' imo.