Why would maximum fly-time (endurance) not coincide with max L/D operating point?
Maximum dwell time or maximum endurance occurs when the power required is minimum. Hence, in this case, the maximum endurance speed is one where the power required is minimum, while in case of maximum range speed, the thrust required is minimum.

For maximum endurance, we must minimize the fuel consumed per unit time i.e. the fuel flow. For maximum range, we must minimize the fuel used per unit distance traveled.

In case of propeller aircraft, the fuel flow rate is proportional to the power produced. Hence, the maximum endurance occurs at a point where the power is minimum. For (turbo)jets, the minimum fuel flow occurs when the thrust is minimum. Hence the maximum endurance occurs when the L/D is maximum. For turbofans, it is somewhere in-between.



Consider a propeller aircraft in a steady, level flight. For determining the condition where the energy expenditure is minimum, we have,

$P = W (\frac{C_{D}}{C_{L}})V$

is minimum. For steady flight, we have,

$V = \sqrt{\frac{W}{\frac{1}{2} \rho S C_{L}}}$ 

This gives,

$P = \sqrt{\frac{W}{\frac{1}{2} \rho S}}(\frac{C_{D}}{C_{L}^{\frac{3}{2}}})$

Thus, for propeller aircraft, the minimum power and maximum endurance occurs when $\frac{C_{L}^{\frac{3}{2}}}{C_{D}}$, rather than $\frac{C_{L}}{C_{D}}$ is maximum. Due to this, the minimum power (maximum endurance) condition occurs at a speed which is 76% of the minimum drag (maximum range) condition. 



Image from eaa1000.av.org

Also, see here and here



Thrust is a force that moves the aircraft. In steady, level flight, this is equal to the drag (if it is more/less, the aircraft will accelerate/decelerate). Power is the rate of doing work i.e. the energy consumed per unit time or rate of energy expenditure (by the a/c powerplant). This is why we are considering minimum power i.e. rate of energy expenditure for determining endurance. 

Power is the product of force (thrust) and velocity. Think of it in this way- as speed increases, the drag decreases, reaches a minimum and then increases. However, as the power is product of drag (i.e. thrust) and velocity, it too follows the a similar path; however, the minimum is reached before the minimum drag. That speed gives the maximum endurance.

For jet engined aircraft, the speeds are different. In this case, the speed corresponding to minimum $\frac{C_{L}}{C_{D}}$ gives maximum endurance, while the speed corresponding to $\frac{C_{L}^{\frac{1}{2}}}{C_{D}}$ gives maximum range. Also, see here