There are many points of disagreement between Fatah and Hamas; so many that they fought an ugly civil war in 2007, leaving Hamas in control of Gaza and Fatah in control of the West Bank. It is a mistake, however, to conclude from their often violent enmity, that Fatah, the so-called "moderate" faction, is or can be a partner to Israel in "peacemaking" or in finding the "two state solution" so beloved of Western politicians.
It is also a mistake for the U.S. and the West to push Israel toward concessions to Mahmoud Abbas in the hope of strengthening Fatah against Hamas.
The enemy of my enemy is not my friend. It is entirely possible for two parties to hate each other, but to agree they hate you more. And so it is in this case. Hamas and Fatah are not opposite ends of some mythical Palestinian political spectrum – they are merely different approaches to the same end.
Hamas is a branch of the Muslim Brotherhood, rooted in Sunni expansionism but aligned with Iran for purposes of money, training and weapons.
This is another instance in which two parties (Sunnis and Shiites) can be at war at one level, but agree to make war together on a third party (Israel). Fatah is open to a (very temporary) political settlement with Israel as long as it brings millions of Arabs into Israel over whom Israel would exercise no functional control.
For both Fatah and Hamas, the bottom line is that the establishment of Israel in 1948, with the blessing of the United Nations, was a mistake by the international community that needs to be corrected.