In Parshat Balak, Balak tells Bilaam that הִנֵּה עַם יָצָא מִמִּצְרַיִם הִנֵּה כִסָּה אֶת עֵין הָאָרֶץ, however, when Bilaam is repeating this to Hashem, he says הִנֵּה הָעָם הַיֹּצֵא מִמִּצְרַיִם וַיְכַס אֶת עֵין הָאָרֶץ. Bilaam is calling Bnei Yisrael the nation as oppose to what Balak said, which is just a nation. Rashi gives explanations for all the other differences between the wording of Balak and the wording of Bilaam, but he doesn't mention the reason for this subtle change. Why does Bilaam say this?
Rav Moshe Feinstein in Darash Moshe said that Balak thought that this was like any other people who had broken free and was now searching for a place to conquer and call home. As a result, he wanted to keep them away from his area as once they settled somewhere, they would assimilate and be a normal nation.
Bil'am knew that as we have it in the haggada, the Bnai Yisrael would never forget what Hashem had done and could wind up affecting all the rest of the world. This was something he definitely did not want to happen
According to Rav Hirsch, Balak said the main situation was not that this was a particular people, but that a people who had been completely submerged in the power of the Egyptian Empire had wrested themselves free and were now going to completely destroy the surrounding nations (even if they are not going to attack Moav directly).
הִנֵּה עַם יָצָא מִמִּצְרַיִם The repetition of הִנֵּה shows that with עַם יָצָא מִמִּצְרַיִם a separate thought is given for his consideration. The fact that עַם יָצָא מִמִּצְרַיִם, that a mass of people politically and socially submerged in the power and nationality of Egypt could go out of Egypt as a socially united people to freedom and independence, marks this people as something unique. And it is to the cause which could have effected this remarkable unheard-of event which your mind must concentrate on, if you are willing to help my purpose to be achieved.
On the other hand, Bil'am knew who they were and what was occurring as a result of Hashem's specific will. He knew that it was not some random people showing the breakout from Egypt, but was the people, the specific people that Hashem had taken out גוי מקרב גוי and the answer to Balak's request as asked would be Tell Balak not to worry, they are moving at my explicit command and will not invade you. Bil'am, on the other hand wanted to see them destroyed and actually tried to get Balak to fight them. As Rashi says on Balak 22:11
curse it: Heb. קָבָה לּי. [This expression used by Balaam] is stronger than אָרָה לּי [used by Balak in verse 6], for it specifies and details [the curse]- [Mid. Tanchuma Balak 5, Num. Rabbah 20:9]
and drive it out: of the world. Balak said only, “and I will drive him out of the land” (verse 6). [His intention was:] I want only to get them away from me, but Balaam hated them more than did Balak. — [Mid. Tanchuma Balak 5, Num. Rabbah 20:9]
Thus, Bil'am referred to this people in order to avoid any possibility of it being someone or something other than Bnai Yisrael as well as to avoid the possibility of being told to reassure Balak that they would not invade Moav and would not need to be fought off.
Malbim (Numbers 22:5) explains that Bil'am's description of "the nation" is more natural, since the Jewish people and their exploits were well known. Balak, on the other hand, tried to downplay Israel's fame and greatness, referring to them as merely "a nation", so that Bil'am wouldn't be discouraged from confronting them.
Alternatively, perhaps the changes are inconsequential. Perhaps, for example, Bil'am didn't remember the exact wording. That the changes are inconsequential is implied by the Akedat Yitshak (Shaar 82) who writes that Bilam and Balak said the exact same thing:
ויאמר בלעם וכו' הנה העם היוצא ממצרים וגו'. עם היות שהמאמרים הללו הן הן דברי בלק