Dame filling is important in Japanese rules. As @_KoBa mentioned, filling dame can compel your opponent to play moves within their own territory, which affects the score (teire).
Technically, under Japanese rules, fill-able dames should be played in order to properly determine whether or not stones are alive in seki. Determining whether stones are in seki is important, since Japanese rules does not count “eyes in seki” as territory. Here are the relevant rules:
From Article 8 of the Translated Japanese Rules
Empty points surrounded by the live stones of just one player are called “eye points.” Other empty points are called “dame.” Stones which are alive but possess dame are said to be in “seki.” Eye points surrounded by stones that are alive but not in seki are called “territory,” each eye point counting as one point of territory.
Thus, if dame are not filled, one could claim that the adjacent stones are only alive in seki and that there eye points do not count as territory. Of course, that’s not what actually happens in practice with reasonable players, but I think the rules are there for particularly tricky life/death/seki situations.