Artificial Intelligence is whatever it is about computing that is sort of magic. As users we don't know why exactly it works but it just does. As builders it's like the dumbest of magic tricks: fast hands, misdirection, brute force beforehand. Sure a lot of research has gone into clever math for it, but once you look behind the curtain, you realize the excitement of the builder is in fooling the user, passing the Turing test by whatever means necessary. (I exaggerate considerably for effect. There's lots of rocket science behind the curtain, but what distinguishes it from algorithms is that it embraces inexact heuristics rather than shunning them).
AI doesn't have to be electronic. It could be mechanical, like a ball-bearing finite state machine that computes divisibility by three, or biological (like a chicken taught to do tictactoe). The chicken winning is magic. The brute force is determining the tictactoe decision tree then teaching the poor chicken. In the end, AI is mostly just computers.
It is usually something that humans are only able to do: language, vision, and logic.
AI is often used for just a heuristic such as the game 2048. The AI used to try to get a better score is essentially heuristics found by a good human player that were then coded as rules in a deterministic program. When you open the hood, there's no rocket science, it's just "always make a move that keeps the highest item in the corner", a human thought that is better than random and better than a beginner, but not perfect. An airline flight suggester is just (OK it is sorta rocket science) a special linear optimization problem (on a lot of data). Linear optimization is usually not considered AI but let's not quibble.
AI usually means that some learning was involved at some point but usually that learning is not continuous, learning in operation. Probably some ML algorithm was run on a lot of past data to generate a rule and then set in stone (until another pass on more recent data updates the rule).
The short history of AI is that it was invented/named in the late 50's, expected to solve all problems in a couple years in the mid 60's
Cognitive computing is IBM's way of reintroducing AI to consumers. It's not AI, but it's not not AI. That is, it is AI dressed up as usable applications or modules that can be fit together to give the appearance that a human is behind it without having an actual human having to step in to do it. The usual list of properties that a cognitive computing app has are: awareness of context of the user, giving the user what they want before they ask for it, they learn from experience, deals well with ambiguity. But then it will probably also incorporate human language input or visual pattern recognition or thinking through a number of inference steps.
Do you need an app that'll give you a new good tasting recipe for tacos? Deciding what's best is probably a good human task (but shhh we have an ML algorithm that figured out what are good ingredient combinations). Do you need an app that suggests to you good personalized travel plans? And now for something actually practical, do you need an app that will help discover cancer cures from buckets of EHR data?
All of these are not the usual single narrow one-off AI apps (back up a trailer, distinguish cats from dogs in images, compete in rock paper scissor competitions).
Without a doubt, cognitive computing is totally a hype/marketing term, new enough not to be some old over-used baggage-laden term like AI, not misleading (these are all sort of thinking apps) and vague enough to allow all sorts of companies to jump on the bandwagon with "Why yes, we've been doing cognitive computing before the term was invented!"
But hype terms can be useful. Cognitive computing is a good label for engineering a combination of features, some that are traditional AI and some that are just good design that are becoming more obvious to have. Whether it's the machines doing the thinking in silicon, or the engineers doing some extra thinking in design, as long as the machine looks like it's reading your mind then that's a good app. It'll be useful if it catches on.
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