Both are quasi-experimental designs used to measure the effect of some running variable near the cutoff value. Bunching is becoming popular in economics, regression discontinuity design (RDD) is more or less a common tool in social sciences. Are both of them the same, just with different maths, or they have some fundamental differences?
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Both methods are certainly related. The main difference is how the related estimators are obtained. Bunching Estimators calculate the deviation of the density at the point where bunching is expected to occur. The deviation is defined by extrapolating the density without the bunching area. One might call this an artificial control.
In RDD settings, you (should) have good theoretical reasons to assume that observations just left of the discontinuity are a good control group of the observations just right of it (or vice versa). In RDD settings, one can hence observe the control group directly.
