How much of an advantage is it to really specialize your cities?
Over the course of a game, about 10% more production, empire-wide. City specialization is all about reducing wasted production turns. Another way to think of it is maximizing the turns-to-benefit ratio, when discussing turns of production. Every city should get happiness boosters so that it can work more tiles, but other buildings should be decided upon according to the city's specialization. The main categories that most people recognize are science, production and GPP.
If you have a city whose terrain only realistically allows you to work two cottage improvements for most of the game, then having that city spend time building research boosters is of so little benefit that you might as well call those wasted turns. Those turns could have been spent producing units. Likewise, in a city that has very few hammers, building production-boosting buildings is a waste of time. The benefit that you get in return will be so small that the many turns spent building them can be considered wasted. Instead, build science-boosting buildings and then either build research or wealth. It might seem complicated at first, but it becomes second nature once you get used to the concept.
On most maps you'll have some city sites that are clearly, unequivocally suited for one category of specialization. Other areas will result in what are called "hybrid" cities - basically cities that are pretty good at research and also pretty good at production. In those areas you don't specialize because the city can make good use of just about any building.
I found that an Oasis map has terrain that is really well suited to learning city specialization. Some areas will naturally lend themselves to production cities, while others will naturally become science cities. If you play a couple of games on an Oasis map then you should get the hang of it pretty quickly.
Of course, you don't need to specialize your cities to have fun or even to win, but specializing will give you a slight advantage that you wouldn't otherwise have. When you consider the bonuses that the AI gets when you move from one difficulty level to another, city specialization basically gives you enough of a boost to let you step up the scale one notch. City specialization might sound like a lot of work, but it really isn't. It isn't micromanagement. It is simply common sense in terms of what is useful and what is wasteful in terms of production. Once you get used to it, you won't even consciously think about it. It will become second nature, affecting your production decisions without conscious effort on your part.
FfH has so many neat buildings that it can be difficult to resist the temptation to build the ones a city doesn't really need. Focusing a city's purpose (and therefore its build orders) will improve your investment-to-return ratio, though, and is a wise choice. If you want to specialize, it will help you just enough to bump the difficulty level up one notch. If you don't want to worry about it, though, then don't. It helps but isn't necessary for a fun game. It is simply an option to consider.