In general, the astronauts could operate the spacecraft autonomously without help from mission control. Specifically, you could land on the far side of the moon.
The Apollo spacecraft each had a guidance computer, but these were considerably limited in program memory and processing speed. The difficult calculations were performed on IBM mainframes back in Houston. Prior to major operations such as injections, landing, and rendezvous, mission control would radio up to several dozen parameters that the guidance computers needed. These were called "PADs", and you can find them throughout the Apollo flight transcripts.
There seems to be a frequent misconception here that the Apollo spacecraft could be operated autonomously, either by the guidance computers themselves, by the astronauts calculating their own numbers, or by the astronauts flying manually. None of these were at all possible. In fact, for many years Von Braun and other officials pushed the "direct" approach with a single spacecraft making the whole trip, arguing that there was no way that a lander ascending from the lunar surface could ever rendezvous with a spacecraft in lunar orbit. Quoting an interview with Robert Gilruth, the first director of the MSC in Houston:
DeVorkin: In direct descent you needed an enormous booster. In earth orbit rendezvous, you needed two Saturn launchers to meet in orbit. In lunar orbit rendezvous, you needed only one Saturn launcher, but you had to have, correct me if I'm wrong, extremely finely tuned abilities to do celestial navigation, because the lunar orbit rendezvous was being done at the greatest distance, was the critical path. The most difficult thing to conquer.
Gilruth: But that had onboard navigation.
DeVorkin: Had it been developed yet? To what degree were the computers ready and available?
Gilruth: Well, that's true, we were the people that made IBM. There's no question about it. We put the computer age ahead ten years with Apollo, because we really did use IBM and built them up in order to do this program.
DeVorkin: Let's go back and talk about your comment about IBM, and how NASA made IBM what it is today.
Gilruth: I think I would say that they had a lot of talent. They would have become successful no matter what, but we did help them by giving them such a challenging project as Apollo was, which required the utmost in computer development. I'm not a computer expert, although I had some very good people in that work. Without those computers, we never could have solved all those equations in such short time, that we could direct these things into proper orbits.
It wasn't until the Gemini program proved that rendezvous was possible under computer control that the Apollo program switched to the lunar-orbit-rendezvous method. But it wasn't the on-board guidance computers coming up with the difficult calculations, but rather the IBM mainframes back in Houston.
Your smartphone has more memory and processing power than those 50-year-old mainframes, and could easily do those calculations. Possibly even a Shuttle-era AP-101 computer could do it as well. This means that should communication be lost with mission control, the astronauts would have a good chance making it back home anyway.
You'd still want mission control. They provide expertise and perspectives that a crew may not have. They also track the position and velocity of the spacecraft, which is absolutely essential and not something the spacecraft can do precisely by itself. However, three TDRS-style satellites in orbit around the moon would be sufficient.
Having autonomous control of the spacecraft opens up opportunities that weren't possible in Apollo. In particular, it would be safe to land astronauts on the far side of the moon. It appears that the Chinese might just do that.