When Roy's message is sent to Neptune, a 'top secret laser' is used to do it. At the speed of light the message would have taken >4 hours to reach Neptune. Roy's father's reply would have taken another >4 hours to reach them, summing a total of >9 hours. But Roy is still inside the recording studio when he asks if they have succeed. It is unlikely he was so much time waiting.
Errors in Roy's free fall at the beginning of the film: Roy should have gone "full tilt delta" (arms and legs fully extended) to increase his horizontal distance from the satellite in order to avoid the falling debris. Once his canopy was damaged by debris, he should release it and deploy his spare.
Cepheus can make the journey from Mars to Neptune in 79 days which means it must have monstrously powerful engines (the same journey took Voyager 12 years). If it has all this power, plus enough spare to detour, de-accelerate, and re-accelerate on a rescue mission, then there is no need for the array of solar panels it sports.
The distance from Tycho crater (where Roy lands on the Moon) to the center of Farside (where the "Cepheus" is located) is roughly 1,700 miles. Traveling at the speed shown and assuming no stops, Roy's lunar rover would take a couple of days to get there.
A nuclear explosion would not provide any sort of propulsion to a spaceship, even if it also caused an antimatter explosion. There is no atmosphere in space and no "shockwaves" that could propel a ship. Only whatever directly hit the ship would provide any sort of propulsion, and it would be a one-time, instantly-over event.
Antimatter, when it comes into contact with normal matter, does not cause a "chain reaction", as is stated when Roy is being briefed about the Lima Project. When antimatter and matter come into contact, they mutually annihilate each other into pure energy, using Einstein's E=mc2 equation (100% matter-to-energy conversion). There would be no possibility of an antimatter incident at Neptune affecting Earth, except for the one-time burst of energy that such an explosion would create.
During the lunar rover journey, dust is seen "hanging in the air". This would not occur on the Moon, as there is virtually no atmosphere; all dust thrown up from the ground would immediately fall back to the ground and stay there.
The speeds at which the spacecraft Pitt pilots to Neptune requires a constant acceleration to make the trip in 79 days. Every external shot of the ship shows its thrusters firing creating this acceleration. This would also facilitate perceived gravity on the ship; however, it is pointed out that the ship has zero gravity.
While landing on Mars after the system failure, manual mode is switched and Roy tries to set the angle of fall to 0, but he doesn't turn off the engines at any time; he succeeds in setting the angle in the very last second and yet the rocket lands right in the place where it should, instead of miles away from it.
The astronaut uses the rotation of the radar, which he sits on, to shoot him off into space with such accuracy that he arrives exactly at his space ship, clearly several miles away. Such accuracy is ridiculous for a slingshot from a rotating beam. In addition, a flat-plane radar would be pretty useless in space.
When Roy is speaking to his father inside the spaceship he is seen floating weightless. Therefore the tear he sheds should pool in front of his cornea (consequently distorting his vision). If there are enough tears and he agitates them by blinking or brushing them away, the excess fluid would float into the air rather than down his cheek.
There was no reason to send Roy to Mars to make a voice transmission to Neptune. Even if it needed to be transmitted from Mars, he could have recorded it on Earth, transmitted it to Mars, then retransmitted it from Mars to Neptune. Likewise, when they asked him to come back to repeat the transmission, they could have recorded the first transmission and retransmitted it.
The goof items below may give away important plot points.
Roy uses his spacesuit's reactor jets to launch himself too quickly on final approach to Cepheus rather than also using them to slow his approach -- thus requiring him to make the high-risk manual grab as he crashes into it. This is basic spaceflight physics and since Roy demonstrated his knowledge of this when approaching the Lima Project in his shuttle, it makes no sense that he would make this elementary mistake on his return.
When leaving Mars, Roy sees the huge engines ignite a few feet away from him before he gets inside the rocket, but is not injured. Also, he enters the rocket through a hatch near the engines, but ends up in the crew compartment about 100 meters up.