What's the difference? Is anyone able to cast arcane spells that is not an "arcane spellcaster?" Or does this difference arise simply from different wording in different books?
In short, "arcane spellcaster" and "can cast arcane spells" are functionally identical.
In order to be considered an arcane spellcaster, you must be able to cast arcane spells. Likewise, if you can cast arcane spells, you are an arcane spellcaster. There were rules in early 3.5 that classified creatures with spell-like abilities as arcane for certain purposes, but that was errata'd out.
In reality, neither term is explicitly defined
Both terms were used expecting that players could “figure out” when they apply, and in most cases they are both rather clear and amount to the same thing. It’s clear that a Wizard is both “an arcane spellcaster” and is “able to cast arcane spells.”
Unfortunately, 3.x has an absolutely enormous amount of material available and some weird edge cases do exist, and it would be nice if we had been given an explicit definition. We were not.1
For example,2 the Southern Magician feat from Races of Faerûn says that
Once per day per two spellcaster levels, you can cast a divine spell as an arcane spell, or vice versa.
Does this make you a “divine spellcaster” if you were a Wizard and took it? Probably not. But do you count as “cast[ing] divine spells” when you use this feat? It certainly seems like you do. But it does say that “the actual source of the spell's power doesn't change, nor does its means of preparation,” what significance, if any, does that have on the question?
The Dragon vol. 325 feat Alternate Source Spell removes the daily restriction on this, and lets you prepare any spell as arcane or divine. Does that change anything about the above? Do you even qualify (“Prerequisite: Ability to cast both divine and arcane spells.”)?
None of this is well-defined by the game. The conclusions of most char-op’ers (who do have very close readings of the rules, even if they are always interpreted favorably) was that it seems to work. Their argument is basically that you can cast spells of the other type, and therefore are a spellcaster of that sort. But the rules never explicitly say so; they just use these terms without actually defining them.
Ultimately, Southern Magician and/or Alternate Source Spell allow for a number of tricks, some of which are cool, flavorful, and not especially powerful (a small benefit for Sha’ir, from Dragon Compendium, that basically allows them to ignore Arcane Spell Failure), and others which very much are powerful (Dweormerkeeper from Complete Divine’s web supplement is incredibly powerful, for instance, and it is much easier to enter using these feats than it otherwise would be).
It only makes sense, in my mind, to bring up these kinds of corner-cases with the DM. Using these combinations is perilously close to the age-old “the rules don’t say I can’t!” line of reasoning that is so frequently frowned-upon. As a DM, I wouldn’t even bother paying attention to the rules much here, but rather on the specific combo being utilized: I’d allow a Sha’ir to use it, would not allow a Dweormerkeeper to, and so on. I think a case-by-case basis, based pretty much solely on the result rather than the means, makes sense in these weird cases.
Most of the time, of course, that’s not necessary because it’s pretty clear.
1 I have looked to see if Pathfinder provided such a definition; I could not find one. I can say for sure that 3.5 did not have one.
2 This example comes from 3.5 rather than Pathfinder, because I am more familiar with 3.5; even if no similar examples exist in Pathfinder at the moment, they may someday and this is mostly for illustration.
Spellcasters with too low of ability scores cannot use spells. They are still an arcane/divine spellcaster just they cannot cast the spells.
An example of this would be a Spellcaster with a 16 Int. He can cast up to 6th level spells (if he knows them and is high enough level to; 16 is the minimum ability score for that level of spell). He has the ability to cast arcane spells. However if he drops to 9 Int permanently then he can no longer cast any of his spells; he knows them he just cannot cast them. (If it were temporary his ability to cast spells would be merely disabled.) So he is therefore an arcane spellcaster who cannot cast spells.