I love real ale, can't stand (most) craft beer. I am not certain of the exact differences, I think the definition of "real" ale is that there's live yeast in the barrel. Craft beers are more inert, there were variations on ale with no live yeast prior to the invention and proliferation of "craft beer" but the "craft" variant differs in taste. I don't know whether it is the type of hops, or the quantity or other ingredients but I find some of these - the IPAs in particular, absolutely repugnant. Even the smell of some makes me faintly nauseated, I've been sick (and sometimes the other end) after just one pint or even a small can of some types. I suspect I might be allergic to something in them. A couple of friends have reported similar reactions.
One of the defining features of most craft beer is that there's often a floral, citrus-y or grapefruit-y taste and smell as
@Sabresonic mentioned above. These ones, usually in the IPAs are the worst for me, though conversely some ales with similar characteristics are fine. I've had a couple of "sour beers" which certainly lived up to their name. On the other hand I actually really like some of the odd flavoured stouts and porters- hints of vanilla, blueberry, strawberry, peanut butter and marshmallow even.
I will also add I don't welcome the price hike with these beers, also mentioned above, nor the hipster element. Brewdog are by the far biggest company in this "game" I think they helped spearhead the name "craft beer/brewery" when they got too big to market themselves as a microbrewery. Their pubs are pretty soulless inside, though this hardly unique to them in the modern day.