This is an interesting and often overlooked political division within the Arab World, where most political conflicts are traditionally classified as Sunni vs Shia, conservative vs reformist, secular vs Islamist, or pro-Western vs anti-Western. To answer you question succinctly, it was a result of American imperial power eclipsing British imperial power in the Middle East.

As you point out there is a historical grievance between the Hashemites and Saudis, with the former being expelled from rule in Arabia by the later in 1924-25. The realpolitik answer to you question is that foreign patronage played the key role in subsequent peaceful relations.

Although they had been a player in the area beforehand the Hashemites rose to political dominance in Northern Arabia as a result of having sided with the British against the Ottomans in WW1, as made famous in the English-speaking world through the first-hand experiences and writings of T.E. Lawrence. Following WW1 the Hashemites were set up as the rulers of Arabia and Iraq with British aid.

What is less well known in the West is how the Hashemites were forced out of Arabia by the Saudis in the 1920s, and how the British subsequently placated the Arabian branch of the Hashemites by creating the Kingdom of Jordan for them. The Saudis, who even prior to their conquest of the Hejaz had based their authority on their House's close connection to hard-line Wahabism characterized the Hashemites as puppets of the British, claiming that they were not 'true' Muslims and were therefore unfit to rule and defend the Holy Places of Islam, legitimizing their own rule in the process.

The discovery of oil in Saudi Arabia in the 1930s changed the ruling dynasty's view of the West to a great degree, even if they maintained their austere religious views, and with the founding of Aramco the Saudi's firmly hitched themselves first to American economic interests, and by the time of the Cold War to US political interests.

While the British remained a strong independent player the Jordanians would remain hostile to the Saudis, and it just so happened that the two decades of sour relations after their expulsion from Arabia corresponded with the last days of British imperial power. However, after WW2 and Indian and Israeli Independence Jordan found that they could not rely on the UK politically, economically and militarily as they had in this past. This, compounded by British involvement in the disastrous Suez Crisis and combined with the rise of republican and socialist regimes (and eventually modern Political Islam) in the post-WW2 Arab World caused Jordan to increasingly rely on the United States (the Saudi's key ally), and see their interests align with other conservative monarchs like the Shah of Iran (another ruler who was turning from British to American patronage at the time) and their erstwhile enemy the Saudi King. Since that time, and despite dynastic differences and the relative liberalism of Jordan compared to Saudis, relations between the two states have been peaceful.