What is local-NMF? How is it better than original NMF?
The "parts-based" is explicitly stated in the text to be a synonym to "non-subtractive". It is a natural feature of NMF. Its underlying assumption is that all elements of a source matrix are non-negative, as well as that the source matrix is a product of two other matrices, all elements of which are non-negative. Thus the source matrix is a sum of independent "parts". If the parts were allowed to be negative, the problem would lose capacity for unique solutions, since then, for example one could generate out of thin air fake components that are equal and opposite.

The "local" refers to spatial locality. Especially in image processing, frequently each component is expected to be concentrated within a small region, such that all related pixels are not too far from each other. This is not achieved by naive implementation of NMF, which results in components "smeared" across the entire domain, being obviously non-local.