You must understand what the terms "derated thrust" and reduced thrust" mean.
Derated thrust means using a lower limit thrust, i.e. not pushing the T/L so far forward and treating the lower T/L angle as a limit. Hence lower Vmcg and Vmca as well as less wear and tear. In the event of any problem, simply pushing the T/L further forward is forbidden because, inter alia, all speed margins are calculated on a lower max thrust and control may be compromised.
Reduced thrust means setting less thrust than the declared limit thrust when a runway is not limiting for the planned TOW on the day. In the event of a problem it is permissible to push the T/L forward to gain more thrust, although this is generally unnecessary and indeed foolhardy because all certified performance parameters can be met at the set thrust, so why destabilise the situation?
Practically speaking, you "derate" thrust to, on Boeings, RTG 1 or RTG 2 depending on your margins available after planned TOW is calculated. After derating often it is possible to reduce thrust further using "assumed temperature" or another approved method of reducing thrust if less thrust will cover your planned TOW. Hence there is no impact on range in the big Boeings, just a way to reduce wear and tear.
Less noise is actually untrue - the aircraft generating said noise is at a lower altitude climbing less rapidly when any less than max thrust is set. Look at noise abatement procedures ex LHR.