So why do it this way? Suppose that there is noise on top of the input voltage. By integrating, we average out the noise, thus getting a precise measurement of the input potential. Additionally, measuring time precisely is easier than measuring potential precisely. We can get additional bits in our measurement by using a faster clock (and thus a higher count for the same discharge time) much more easily than by adding bits to a flash or successive approximation converter. For the same mean as in the sketch above, a noisy signal would be digitized thus:

The zig-zag increase in potential is due to noise on the input signal, but that averages out over time T.