After becoming the ruler of Transjordan, King Abdullah made no secret of his ambition to unite the central Arab lands of Greater Syria, which included Palestine, Syria, Lebanon, and Jordan. The ultimate object of his desire was a throne in Damascus. His brother, Faysal bin Husayn, had been elected King of (greater) Syria by the Arab Grand Committee at Damascus on 8 th March 1920, having taken the city during the Great Arab Revolt of WWI. Only months later, the French crushed the emerging Syrian state at the Battle of Maysalun on 23 July and redrew its borders. Driven from Syria, the Hashemites, supported by many Syrians, continued to champion the legitimacy of Greater Syria. Abdullah assumed leadership of the cause when his brother Faysal, having become King of Iraq, died in 1933. No sooner did Syria and Jordan gain independence in 1946, than Abdullah called for the immediate unification of Syria under his crown, believing he could win not only British support for his plan, but also the broad masses of Arabs who were calling for unity.
Shukri al-Quwwatli had good reason to fear Abdullah’s Greater Syria Plan. His own army was small, badly trained, and unreliable. To make matters worse, the president suspected that many of his top officers had either been in contact with King Abdullah himself or his agents in Syria. The Jabal Druze, positioned on the Syrian-Jordanian border, was in full revolt against Damascus by the fall of 1947, having defied Quwwatli following his annulment of the summer parliamentary elections in the province. Its Atrash leaders, renowned for their military prowess, were threatening to secede to Jordan and had positioned themselves to act as the bridgehead for a Jordanian strike on Damascus. King Abdullah sounded out the Druze in both Syria and Lebanon about unifying their regions and giving them a large measure of autonomy within a Greater Syria in exchange for their support in helping to create it.
A border incident could easily give Abdullah the pretext he needed for a move on Damascus . In contrast to Syria, Jordan was a stable state. Its army, the Arab Legion commanded by British officers, was by all accounts the best trained and “by far the most loyal and efficient” fighting force among the Arab League states. Its commander, General John B. Glubb, or Glubb Pasha as he was known, always favored the notion of a Greater Syria acting as the centerpiece of British policy in the Middle East. As he explained to the British government: “It is not fanciful to imagine the Arab Legion as the nucleus of the Army of Greater Syria in the future.” The Prime Minister of Egypt, Nuqrashi Pasha, also acknowledged that the Jordanian army was superior to either the Egyptian or Syrian armies when he proposed to the Arab League in October 1947 that they pay for the Jordanian army to serve as the guardian of Palestine.
Source here:
http://joshualandis.oucreate.com//Syria_1948.htm
By 1947, Abdullah had formed an alliance with Iraq and Turkey against Syria in his goal to conquer Syria and establish himself in Damascus. So let's say he manages to go through with his plan.
I don't think think there'd be much of a difference in the war in Palestine as the Jordanians were the only really formidable arab military and they would still take the West Bank.
What if Abdullah had managed to conquer Syria and establish a greater Syrian state comprised of today's Jordan, Syria, and (after 1948) the West Bank?