Part 1 (scroll to my reply to this comment to read Part 2):

The Jordanian motivations go much deeper than the history of the Hussein family and the Palestinians. Instead, one has to first understand the development of Jordan and the development of the state of Israel to understand their overlap.

On the one hand, Jordan has participated in a number of conflicts against Israel, including joining the efforts to destroy it in 1948, fighting alongside Arab armies in 1967, and participating in anti-Israel actions in various international forums. At the same time, as you noted correctly, Jordan has been far friendlier to Israel than other Arab states, going so far as to propose ways to avoid war in 1948 (King Abdullah proposing in 1947, for example, a nonbelligerency pact to split the Mandate) and to allegedly even warn Israel of an impending attack in 1973 (King Hussein told Golda Meir in a secret meeting on September 25, <2 weeks before war, that Syria's military had moved to pre-war positions). Evidently, the willingness to cooperate with Zionist leaderships has extended to before King Hussein's father (King Abdullah I) was assassinated by a Palestinian fearing peace with Israel.

The roots are both strategic and also demographic. Through a variety of factors, Jordanian leaders have been forced to confront that in many ways, their goals overlapped with those of Israel historically.

Jordan, for example, was just as concerned as many other Arab leaders with the prospect of taking over the Arab areas of the British Mandate. However, unlike the other parties, Jordan was also party to the best-trained military in the lead-up to the 1948 war. Its well-trained military was counterbalanced, relative to the other Arab states, by the fact that it was (and has been) the smallest country population-wise of the major Arab belligerents since 1948, meaning it also was the least competitive in gross resources to bring to bear. This meant that Jordan was far more willing than the other parties to conflicts with Israel to accept that a secret compromise might net them better results (i.e. an agreement to gain the Arab portions) than to compete with larger Arab neighbors, such as Egypt or Syria. This was compounded in importance by the fact that Egypt and Jordan were rivals in the lead-up to the conflict, and approached it from different perspectives. Jordan, advised by the Arab Legion's head (a British commander named John Bagot Glubb), believed that the Arab forces were naive to think they could defeat the Jewish forces, and Abdullah was most conscious of this, saying that, "The Jews are too strong -- it is a mistake to make war". While other Arab leaders doubted victory, their inept military commands and the Arab League's closest thing to a plan seemed to approach conflict with overly optimistic ideas. As Glubb put it, the war pitch led to a situation where "Doubters were denounced as traitors", but only Jordan appeared to be as clearly aware of just how poor the planning/capability was and therefore willing to seek escape from the strictures of war.

Besides the question of gross resources, competition between Arab states, and awareness of military discrepancies, there existed yet another reason why Jordan and Israel often found themselves drifting together. This reason lay primarily in the shifting demographics of Jordan following the 1948 war. The influx of Palestinian refugees from this war into the West Bank and Jordan, and Jordan's desire to solidify its control over the West Bank, meant Jordan had an inherently aligned strategic interest in integrating Palestinian refugees with Israel's desire to see those refugees integrated (to avoid claims for return to Israel in the long-run, a situation that has persisted to present-day). At the same time, Jordan sought integration in a way to attempt to reduce Palestinian nationalism, and to buffer his own claim to the West Bank, a claim that the Arab world largely never recognized (but that Israel, in 1947, was semi-willing to accept in exchange for peace). While this proposal never included Jerusalem of course, the common ground there was far greater than the common ground with Egypt or Syria, who treated Palestinian refugees quite differently and had gained little territory as a result of the 1948 war (and the land they did gain was of questionable value, compared to the Jordanian gains of coveted Jerusalem and arable land). In fact, it is even arguable that as Jordan had gained most of what it wanted, its remaining conflict with Israel was more related to the question of Arab unity (until 1967, of course) than it was to specific claims it wished to make to Israeli territory. Jordan, rationally, feared Palestinian reactions if it sought to overtly accept Israel's existence (amplified by the assassination of King Abdullah in 1951 predicated on this fear), but it also feared Palestinian reactions if it sought too strongly to erase Palestinian nationalism without care. As a result, Jordan's balancing act largely failed: Palestinians never came to view themselves as true Jordanians, were excluded from Arab Legion combat formations, and were barred from high-ranking positions in the civil bureaucracy, for fear of potential coup (particularly since Palestinians made up such a large proportion of the population). Egypt and Syria, on the other hand, remained largely low on gains from 1948, still coveted and fought with Israel over things like water and passageways (such as the diversions of water from the Sea of Galilee/Jordan River, or conflict over the Gulf of Aqaba). Furthermore, Egypt sought (particularly under Nasser) a role of regional hegemony, and opposition to Israel and the West were a crucial component of Nasser's strategy in mass appeal. His ability to gain influence in other countries was also a source of great concern to Jordan, Israel, and the West, which led them to covertly remain more friendly than they otherwise might have. Nasser's rise thereby made friends out of those who had previously fought, and who had also had less disagreement prior themselves.