The Chizkuni addresses this, and explains that on the first occasion they were not with Yosef for an extended amount of time and would probably not pick up on the fact that he had been crying. However, on the second occasion they would be spending an entire meal together and it was more likely that they would notice he was crying and he therefore had to wash his face.
The Oznayim Latorah explains that this was a far more emotional crying caused by Binyanim's moving explanation of his childrens' names [see Rashi] (or alternatively that he was informed that Yitzchok had passed away) and this necessitated a full washing of his face. [We also see many clues from the pesukim that this crying much more emotional: "And Joseph hastened, for his mercy was stirred toward his brother, and he wanted to weep; so he went into the room and wept there. And he washed his face and came out, and he restrained himself and said, "Serve the food.". We don't find the first time that Yosef had to run out to cry or that he had to control himself afterwards.]