If you consult Deuteronomy 2:9, you will find that Moab was not to be fought with, being descendants of Lot: and therefore, when Israel passed before Oboth, they did not attack Moab; see Deuteronomy 2:11-13. But the Amorites had no such privilege of exemption; and hence became a prey to Israel.
Numbers 21:21-25
21 And Israel sent messengers unto Sihon king of the Amorites, saying,
22 Let me pass through thy land: we will not turn into the fields, or into the vineyards; we will not drink of the waters of the well: but we will go along by the king's high way, until we be past thy borders.
23 And Sihon would not suffer Israel to pass through his border: but Sihon gathered all his people together, and went out against Israel into the wilderness: and he came to Jahaz, and fought against Israel.
24 And Israel smote him with the edge of the sword, and possessed his land from Arnon unto Jabbok, even unto the children of Ammon: for the border of the children of Ammon was strong.
25 And Israel took all these cities: and Israel dwelt in all the cities of the Amorites, in Heshbon, and in all the villagesf thereof.