My answer draws on the ideas that Zod is a trained entity vs Kal'el who is an untrained entity.
Basically, a bit of history. Krypton is a planet with a Red star, because of this, the Kryptonians act and have the physique and abilities of normal humans. This is explained due to the level of radiation the star is giving off. Jor'El knew this about the planet and also knew that Earth has a Yellow star, signifying a different radiation level on the planet. He theorized (or even knew) that his son would be 'like a god' to humans due to his exposure to the different radiation of our Sun.
Judging by the school scene where his x-ray vision starts bugging out that Clark had always had these powers since arriving and has regular freak-outs controlling his abilities and his 'parents' are used to this, signified by his 'mother' knowing how to quickly calm him down enough. This event also appears to have happened to him around age 8, signifying one of the earliest scenes in the timeline.
Shoot ahead 30 years, Clark has obvious control over his abilities, using them without strain and never losing control of his hearing/x-ray vision/etc.
Now we reach Zod and his gang. Assuming Jor'El's knowledge of yellow stars causing Kryptonians to have immense strength and speed was available to military types like Zod, this would explain why they had a bit of a handle on the strength aspect of the abilities. Also the important fact here is that they are trained military personnel meaning they would be able to easily deal with a rapid change in their body in a better state of mind than a scared child would. Remember that in this continuity, Kryptonians were bred for a specific purpose and all 3 of these Kryptonians were bred for war and combat.
Now it also seems that the x-ray vision and eye-beams comes from direct exposure to the atmosphere instead of just being within the radiation. We can only assume that it requires the eyes themselves to become exposed to the open air in order to trigger this change. Clark notices this slight weakness and exploits it in the battle. The only other explanation is that it could be a reaction caused from breathing in unfiltered Earth-air when it reaches their lungs that causes this direct change.
If memory serves, Zod did also have the strength and speed of the other warriors, just got exposed first to the air and retreated.
TLDR: Earth's sun causes Super Strength and Speed, Earth's Atmosphere causes X-ray vision and laser eyes.
