"The rectangle method" is an ambiguous phrase. Is it the midpoint rule, or the right-hand Riemann sum, or what?

Anyway, the two methods will give similar results if the curvature of the function is slight. If the function is concave upwards, the trapezium method will give a higher answer than the midpoint rule; if the function is concave downward, the trapezium method will give a lower answer than the midpoint rule.

One cannot in general say which method gives a "better" answer. For example, in approximating the integral from 0 to 1 of x^2 dx by using n = 1, the midpoint rule gives 1/4, which is a better answer than the 1/2 you would get from the trapezium method. But if you start with a function that has a bell-shaped curve in the middle of it, and use n = 1, the result given by the trapezium method will give a better approximation than the midpoint rule.