Chemically, nicotine affects a much broader range of neurotransmitters than caffeine. Caffeine is pretty much just a stimulant, whereas nicotine can produce a whole range of different effects. The most important thing though is that nicotine is vastly, vastly more addictive. Nicotine actually interferes with the brain's reward system in the same way other addictive drugs do, whereas caffeine just gives you a headache for a bit if you don't have it.
Technically, nicotine affects precisely one neurotransmitter system (acetylcholine), just like caffeine (adenosine) - directly, anyway. The issue is what that neurotransmitter does. Acetylcholine has effects all over the body. There are two types of receptor (a protein that binds the chemical to cause a biological effect) for acetylcholine, termed muscarinic and nicotinic. Nicotine binds to the latter, obviously, and essentially replicates the function of acetylcholine.
Nicotinic acetylcholine receptors largely exist in three places: the neuromuscular junction, autonomic ganglia, and the central nervous system. Nicotine actually binds incredibly poorly to the receptors at the neuromuscular junction, which is why it is a narcotic rather than just a toxin. It would be bad if it bound to receptors at the NMJ effectively: muscle twitches, spasms, paralysis depending on dose.
So, the other two. In the central nervous system, these receptors are all over the place, and the long and short of it is that they increase cellular excitability - make neurones more active, which roughly translates into psychomotor stimulation. They have fairly strong effects on some specific pathways, such as the reward pathway, and hence part of the addictiveness. Also often increases aspects of cognition (memory, concentration, etc.), and dampens appetite.
Autonomic ganglia are part of the control system of organs by the central nervous system - essentially little relays between the spinal output and the organs themselves. Acetylcholine and nicotine activate them - however there's a trick here, because the autonomic nervous system is split between the sympathetic nervous system ("fight or flight" - think adrenaline which activates this system, although the neurotransmitter is actually noradrenaline), and the parasympathetic nervous system ("rest and digest"). Those two create effectively contrary effects on target organs, and nicotine activates both, so the end result of nicotine on the autonomic ganglia is... complex. However, in practice, a lot of the pleasure of nicotine is also that it causes the release of adrenaline from the adrenal gland. Adrenaline is actually a lot of the "rush" of nicotine. Adrenaline also increases excitability in the central nervous system, and potentially contributes to addictiveness via the reward pathway. Incidentally, the drugs probably most famous for increasing activity in the central nervous system through this pathway are amphetamines (alertness, excitability, and of course distinctly addictive).
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Adenosine is probably more prevalent as a transmitter in the body than nicotine. It has important roles in blood pressure, heart rate and force, and it's ubiquitous in the central nervous system. However, it doesn't really have many very strong effects on specific pathways in the brain in the way nicotine does, which in a simplistic fashion is why it's less problematic. Adenosine mostly decreases neuronal excitability, and caffeine blocks the effect of adenosine, which is why it's a stimulant.
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Nicotine is not very dangerous at normal doses. Not acutely, or chronically. The vast majority of problems associated with nicotine consumption are smoking, not nicotine itself. Nicotine is not good: in the long run it will probably stiffen your arteries and heart muscle, and predispose you to cardiovascular problems. Even then, it's nowhere near as problematic as stuff like cocaine or amphetamines. Nicotine is almost certainly less dangerous to your health than alcohol is. However, it is much more addictive.
We're yet to find out how bad vaping is, but it's surely vastly less dangerous than smoking. Assuming the vape juice is made properly, and there have been cases of crappy forumations by amateurs in the US killing people, because it's been unregulated. Chances are vaping will irritate the lungs and cause increased risk of emphysema, COPD, and some cancer risk. But vastly, vastly, less than smoking. Already alcohol is by far the most damaging drug at a societal level, and even if 100% of the population took up vaping, it wouldn't come close to alcohol. So, overall, I would consign vaping to a matter of very low concern. Although I think children and adults should be dissuaded from it (looking at you, Juul) as a general rule because nicotine is addictive.
A lot of the attacks on vaping - which are extensive as Eacaraxe pointed out - in my view are really a sort of Puritanism and anti-drug sentiment coupled with a suspicion of the new, plus cigarette companies defending their market until they can get their own replacements (e.g. IQOS) going. If vaping gets people off cigarettes, it's great news. And if we end with more people vaping than we now have smokers, no big deal. But Juul... fuck Juul, for trying to kiddies hooked and buying science themselves.