Korn, Soil and Disturbed are pure nu metal. Otep are very hard to label (often called 'Avant Garde Metal'😉, but they are definitely not nu metal, neither are Kittie. Nu metal is generally heavy metal in which the guitar riffs are percussive and without solos, and often contain hip-hop style drums and samples.
Typical nu metal bands:
Korn
Limp Bizkit
POD
Adema
Hardcore is difficult to explain...it used to be the term for the harder punk groups like Circle Jerks, now its used for all kinds of things.
All these bands have been classed as 'hardcore':
The Dillenger Escape Plan (experimental, very fast and heavy)
Glassjaw (screamo type band)
Willhaven (noisecore band, lots of distortion)
Thursday (more along the lines of emo)
Poison The Well (metalcore)
Refused (drift from punk to emo, mostly punk tho)
Cave-In (same as The DEP)
Its very vague and confusing 😖