I realize this is an old question, but I came across it while looking for something else and it bugs me that the existing answers are conjecture and incorrect (though I'm sure well intentioned). So, in case anybody else comes looking some day...
The advantage of the APU in this situation was to provide greater power. When the electrical busses on an A320 lose power, a small windmilling propeller called the Ram Air Turbine (RAT) deploys automatically when the aircraft is above 100 knots. The RAT provides a minimum of power to support crucial aircraft systems, and it also powers one of the three hydraulic systems. The APU, on the other hand, can power the entire aircraft.
As far as the electrical side of things goes, this allowed the aircraft to remain in Normal Law. Airbus aircraft use a set of flight control laws that determine certain protections and operating margins. In this case, it allowed the aircraft to remain a safe margin above stall speed for as long as possible. Without Normal Law protection, which would've been lost if operating only on the RAT, some of those protections would be lost. In other words, it allowed greater control protections to help with a smoother landing.
On the hydraulic side, it gets a little complicated. The three hydraulic systems are variously powered by the engines or electrical supply. The RAT can only power one of these three systems (blue). With this single system powered, some controls would operate normally, some at half speed, and some not at all. The APU can directly power the blue system, same as the RAT, and could power the other two systems depending on some switch settings. The US Air crew didn't have time to change those settings, but as it turns out, they didn't need to. The hydraulic systems weren't leaking, and with the short time between the birdstrikes and forced landing, enough pressure existed to operate the control surfaces normally.
So, short answer: the APU was helpful in that it provided electric power, which enabled the crew to have better control of the aircraft. It didn't affect the hydraulics, which remained pressurized, and it did not add thrust. It was one of many things that the crew did right, but not the only one. How much of an impact it made it difficult to say, but in an emergency scenario, you'll take all the help you can get.
Edit to add: Another user asked for some documentation regarding the APU and hydraulic systems. All aircraft have flight manuals, including airliners, though the more complex they get, the longer the manuals. At the airliner level and on the pilot side, these are usually distributed as a Flight Crew Operating Manual, or FCOM. FCOMs are not typically publicly available, as some of the information inside is proprietary or otherwise sensitive.
With that said, you can find nearly anything via Google, and there are a number of Airbus A320 FCOMs floating around online. Here's a link to a website featuring some generic A320 FCOM sections which are not specific to any particular air carrier. These are typically issued by Airbus for companies to customize or for training purposes. Here's a specific link to the APU section and one to the Hydraulics section (these two are PDFs). Note that in the first, hydraulics are not mentioned among the APU's functions, and in the latter, the APU is not mentioned among the sources of power for the hydraulics system.
Finally, here's a picture of the overhead panel on the A320. One of the nice things about the A320 flight deck is that there is a basic systems diagram on some of the panels, acting as a quick reference for major components of the system. Note how Green and Yellow systems show engine pumps, Blue and Yellow show electric pumps, the PTU connects Green and Yellow, and the RAT can also power the Blue system. Were hydraulic power available from the APU, it would be shown here (and would require a pushbutton), consistent with Airbus design philosophy.