Modern people and Neanderthals were apparently more like different races-they were both part of the human species, according to the DNA. You can believe as you like, or course...
Some tribes of native Americans disappeared entirely, and over a rather short period of time, although their DNA remains because they interbred with people from other tribes and even other races. Unfortunately, they did not interbreed enough, or before they were already declining in fertility because of inbreeding due to isolation from geography, climate, etc-but that did not make them another species-and neither were Neanderthals...