What caused Jesse Pinkman to go spiralling down after killing somebody?

Walt's progression of killing was very different from Jesse's.

Walt managed to kill Emilio with phosphine gas, then strangled Crazy 8 with a bike lock. He tried to poison Tuco but failed, and he then caused Jane to die and watched it happen. Many others died that he was indirectly responsible for, and his brother-in-law Hank was almost killed because of his actions.

All of these deaths can kind of be justified. The first 2 killings were in self defense, as he would have been killed had he not killed them first. He tried to poison Tuco for the same reason, and it could be argued that Jane killed herself, even though it was Walt who put her in that position.

He also maintains distance from all of their deaths. Emilio is on the other side of the RV door to Walt when he dies, Walt is stood behind Crazy 8 as he strangles him, poisoning is a more indirect method of killing (though he never succeeded anyway), and he simply failed to prevent Jane's death. Even though plenty of other people died through his actions, he was never directly involved.

By the time he runs over the 2 gang members and shoots them, he's already made leaps to justify what he has done multiple times, so to go from that point to running over and shooting people at point blank range is less of a drastic move.

Jesse experiences things completely differently.

He is never directly connected to any of the deaths. He simply witnesses the aftermath of them all. He isn't anywhere near when Emilio or Crazy 8 are killed, and he simply thinks Jane killed herself. He needed to get high to even confront the 2 gang members, and then is still saved from doing anything by Walt.

He then goes to Gale's house, looks him in the eye, and shoots a completely defenseless man (who is pleading for his life) right in the face.

Jesse is the only one who is directly responsible for the death, has none of the distance that Walt ever had when he did it, and has barely any justification. It wasn't Gale's fault that he died, it was simply a way to make sure Gustav had no other options.

This is a much more severe progression, so it affects him much more because he has never done anything even close to this. It is a sober, calculated assassination of someone who is completely innocent. This is why it has so much more of an effect on him.

What caused Jesse Pinkman to go spiralling down after killing Gale?

As you said yourself: "Jesse had contemplated killing so many people, of course he did not actually murder anyone himself, but he was a witness to so many murders."

Witnessing a murder and killing someone cold blooded is different thing. Gale begged him for his life in front of him and he could have spared his life. Jesse might be in bad neighborhood but he always shown to have a soft side and doing something this extreme affected him in quite large way.

How come committing a murder did not affect Walter as much as Jesse?

Walt was losing sanity day by day after knowing that he will die from his condition. He is not the innocent teacher he once was. He is a changed man, as much as he becomes an antagonist from a protagonist, that's his journey from sanity to insanity. He still has a soft corner for his family and Jesse, others not so much. But deaths did affect him, refer to this question for it:

What personality traits did Walt pick up from his enemies?

A lot of it is because the drivers. Jessie is along for the ride, Walter is in control.

As Walter goes though the killings, he has nice packaged justifications for them. If I didn't kill them they would have killed me, and so on. Rather right or wrong, (different ends of the scale depending on where you are in the series) Walter, every time accepted that he was going to kill, then killed. He justified his actions. The only time (IIRC) that he felt responsible for a death he didn't plan out exactly, was the plane crash, and he quickly justified that away.

Walter is an intelligent guy, and with intelligence comes the ability to think more and more, with only a "logical" mind set. To set aside the emotions and focus on justification. If I didn't kill so and so I would have died. So me killing so an so is fine. In fact, towards the end he ends up doing that with such ease, that Walter is now a "bad guy" and not a "good guy doing some sorta bad things"

Jessie on the other hand is not as intelligent. He is not the leader of the pack, and is not making the decision to kill ahead of time. When he has been involved with killing it has always been to watch. He is always more emotional about it. He still comes around to the us v.s. them idea, but he is always lead there.

Jessie is constantly being manipulated by Walter. In addition to which he never wanted to be in the situation in the first place. He'd much rather stop. But he can't. Walter keeps pushing him back in.

So then we get to "the" killing. Walter sees it as his only choice, and thus manipulates Jessie into doing the deed. Jessie on the other hand has many choices. He could "tell" on Walter. He could kill Walter. He could run away. He could just go home. He could get so high that he can't do it. But instead he lets him self be pushed into doing the killing.

Where Walter had nice justifications and an internal system for making killing ok, Jessie, had only "I don't want to do that" that was constantly being hammered against by Walter.

When Jessie gave in and killed, he lost control. He give it away. He knows it. That loss of control is the real wrecking ball. Not only did he do this thing he didn't want to do, he did it because someone else made him.

Nice question. It's because the way these two characters are made. Think in a perspective of a person who thinks that he wasted his entire life without doing anything wrong but landed up in a situation of fighting cancer. Nothing seems wrong in his eyes. Walter White , himself admits in last season that all he did was not for family, but for himself. That's how he is. So killing doesn't affect him mentally.

Now think from Jesse's view point. He has a caring tendency towards children and their future. This can be seen in multiple episodes and scenes. He thinks that his life is off the track because of lack of nurture and hope. He doesn't want it to happen to anyone. Well, his life is now mixed up with drugs and drug addicts, still his imaginative wishes reside inside his heart. All these make him a man who can't tolerate killings.

Hope my answer covered your question.