Fixed size type has a size you can rely on. That is just the point of fixed sized type.
So you now are sure that the type you use at one side will always match the type you'll find to the other side.
Note that C object files doesn't bring with them information about type, so you end up with really nasty bugs when the type size mismatch.
Currently, this is a problem for long, mostly, because the long size is really inconsistent across plateforms.
What is the problem of non fixed size type ? Well, they have non fixed size. You can rely on this size. You know thing will not explode in an unexpected way on some plateform.
Many header of SFML are already using types like sf::Uint32 and similars ones. So I guess you already see a point to have known sized types.
This isn't blablabla, all recent languages have known sized types. Call me crazy, but you already did that in several parts of SFML. And I can't see any good reason for most use of long in the public API.
Like, for exemple, the style of a RenderWindow. This can be 64 bits long, but still, those extra 32bits are non exploitables (I would make SFML incompatible for 32 bits or require the use of fixed size type for 32 bits SFML). The same goes for window. The same goes for Text.
The same goes for any usage of non known sized type in the API actually.
The int everywhere as not a problem as long as SFML isn't ported on other plateforms. I did work on plateform where int wasn't 32bits and I can assure you that this can become quite a mess quickly. You don't immagine how much code rely on the fact that int is 32bits (but actually isn't always).
Just try to link SFML with another language. Every missized variable will compile just fine. And then only god knows what happen at runtime because the calling convention is screwed. You don't get any error. You simply segfault or get plein wrong results, sometime you even get correct results that turns wrong with some criteria on input values. And you have actually no clue if your code will work on any plateforms because you cannot be sure of the size of thoses types. Long is especially inconsistent (it isn't 64bits on all plateforms).