Tonic,
Aircraft take-off performance is affected by many things. Aircraft weight, OAT, pressure height, runway available and obstacles to be cleared after becoming airborne.
Engine power basically reduces with increasing temperature.
One type of flex takeoff is the assumed temperature method where after consulting the RTOW (Regulated Take Off Weight charts - usually unique to each individual runway and take off flap setting ) one determines that for the atmospheric conditions on the day, and the weight of the aircraft, that the aircraft can take off.
Further it could take off at say 40 degrees C BUT this would require less than full power but the aircraft would still meet all regulatory performance requirements.
So, to conserve engine power and thus prolong engine life, the take off is performed as if the OAT is 40 degrees (in this example) despite the fact that it is lower.
On the 146s that I fly we can flex up to an assumed 50 degrees C. The limiting takeoff N1 for an ALF502 is up to 97.6% and we can flex back to 88%. The limiting take-off N1 for an LF507 is up to 97% and we can flex back to 89%. Both flex limits are company imposed and I believe more conservative that the flex limits approved by the manufacturer, although both allow signifficant power reductions without compromising safety.
The most signifficant difference is take-off run and subsequent climb to a safe, obstacle clear height. At Ref power (OAT & Pressure height dependant) the take-off run and subsequent climb will be shorter than at flex power.
p.s. We can and do use flex take-off at MTOW