Or HaChaim on Numbers 24:25/אור החיים על במדבר כ״ד:כ״ה

1 ויקם בלעם וילך Bileam rose and went on his way, etc. The only reason the Torah tells us this detail is to inform us that he did not even go to his inn before setting out homeward bound. Nonetheless the Torah had to add the extra words: "he returned to his place" instead of merely saying: "he went to his place," to tell us that he went with his head held high, not like a fugitive. Had the Torah only written: "he arose and went," I would not have known if he went maintaining his usual posture. The Torah also informed us that Bileam went back to his home as he had been told to do by Balak. When the Torah adds that Balak went on his way (at the same time), this is to tell us that he did not see Bileam off.

2It is also possible that the words: "and also Balak went on his way," may indicate that he already lost his position as king at that time as the Moabites no doubt had heard of the failure of Bileam's mission. Under the circumstances, there was no further need for them to have an alien as king over them. This would also account for the fact that he was killed in Midian during the Israelites' punitive campaign against that country.