That's largely due to factors outside of Abdullah's control. Hussein's 47-year reign was defined by the Israeli-Arab conflict and the context of the now-concluded cold war.
After losing the West Bank and Jerusalem in 1967's Six Day War – and seeing the demographics of his nation transformed by a flood of Palestinian refugees – Hussein was a US ally in a region where Arab nationalists like Egypt's Gamal Abdul Nasser were seen as being in the Soviet orbit.
And with his regional standing aided by the 1967 war with Israel (notwithstanding that it was a disaster), Hussein could act as an interlocutor for the US, the Palestinians, regional powers like Iraq and – quietly – the Israelis.
In the late 1960s there were fears his country could be taken over by the Palestinian Liberation Organization (PLO), which he expelled in 1970. He spent much of the 1980s laying the groundwork for his 1994 peace deal with Israel that secured his kingdom and won him lavish US aid. But that largely cleared away the issues that had helped give him an international voice in the first place.
In contrast to his father, Abdullah has little to offer beyond maintaining his country's pro-Western policies. With no natural resources, Jordan carries little economic clout. With 6.2 million people, his country's population is just one-third the size of Cairo's. The US works closely with Jordan’s intelligence agency, but does not need the kingdom as a base for troops.
So while the White House may like Abdullah, there are other factors in their political calculations.