It is written in the Maaseh Rav that the Vilna Gaon said that one is not permitted to cry on Rosh Hashanah. His mekor is the posuk in Nechemia which states, “Al tisablu ve’al tivku…lechu Iichlu Masmanim – Do not mournand do not cry… go eat good food.” Rosh Hashanah is a day of happiness and celebration.
The Arizal, on the other hand, says that every person with a good neshama must cry on Rosh Hashanah. Are they arguing?
Rav Moshe Shternbuch in his teshuvos (2:268) says that they do not argue. The Vilna Gaon is talking about crying from the fear of Yom Hadin and having the overall mindset that Rosh Hashanah is a day of gloom and doom. However, if the tears come automatically from inspiration and longing for Hashem during tefillah, then not only is it permissible, it is praiseworthy. Outside of tefillah, the mood of the day must be happiness and celebration of the Yom Tov.