As shown in Figure 1, BERTScore (precision/recall) can be intuitively viewed as hard alignments (one-to-one) for words in a sentence pair, where each word in one sequence travels to the most semantically similar word in the other sequence.
moverScore goes beyond BERTScore as it relies on soft alignments (manyto-one) and allows to map semantically related words in one sequence to the respective word in the other sequence by solving a constrained optimization problem: finding the minimum effort to transform between two texts.