Bill Vallicella has had some interesting discussions recently (here and here) about actual and potential infinites. I think that the confusions that often arise on these subjects usually come from terminological issues -- that is, 'potential' and 'actual' get used a lot of different ways here. A few thoughts, allowing for the fact that Aristotle's introduction of the distinction is somewhat tangled and in parts difficult to interpret.
Aristotle's terminology is easy to misinterpret; one could take 'potential infinite' to mean something 'not infinite but able to be'. This is a common popular mistake. In Aristotle the term is explicitly connected to 'potential being', and Aristotle does not regard potential being as non-being, but as a particular (and secondary) way of being. That is, 'potential being' is being just as 'actual being' is. Likewise 'potential infinite' is intended to identify a particular way of being infinite. It is not an indefinite finite, but an infinite of 'potential' kind. It is an infinite, in potency.
If you actualized the potential of a potential infinite, you get an actual infinite, the completion of the potential infinite. If the potential infinite were just an indefinite finite, its completion would just be a definite finite, not the actual infinite.
This is all complicated because in English (and several other modern languages) words like 'actual' often have meanings like 'genuine', 'not illusory', 'not fake'. These meanings are not in view in Aristotle's distinction; in the colloquial sense of 'actually', the potential infinite is actually infinite.
In the context of mathematics, we can somewhat crudely put Aristotle's idea like this. A mathematical object is actual when it is constructed by mathematical means, in both whole and part (cf. Met 1051a); if it is not constructed, but merely possible, it is potential, not actual. If we take the two kinds of potential infinite, the one that is infinite by division and the one that is infinite by addition, you could only make them actual infinites if you exhaustively performed the operation that gives them infinite -- division in the case of the divisible infinite and addition in the case of the additive infinite. None of us, when we talk about infinities in mathematics, actually do the complete constructions; we indirectly prove that they must be infinite, and call it a day, leaving the infinite potential. More precisely, for Aristotle, something counts as infinite if it is 'that for which it is always possible to take something beyond it' or 'one thing after another always coming to be'. If you have actually done this for every case, if all of it has come to be', you have an actual infinite; otherwise, you have a potential infinite.
Aristotle famously rejects actual infinites (at least in the proper and unqualified sense of the term); the resolution to Zeno's paradoxes is that the points in them are not actually constructed, and thus not actualized as points, and therefore the infinity is only potential and not actual. But Aristotle doesn't deny that it is infinite. This is why Aristotle says that the infinite in a way is and in a way is not; it's something that he thinks is only found as a kind of potentiality -- but he thinks we genuinely do find cases that require us to recognize that potential as an infinite potential. It has become popular in some parts of academia to consider Aristotle a finitist, but he himself explicitly rejects this by insisting on preserving and using a concept of the infinite even while he is rejecting that any such infinite could be actualized. Indeed, he himself says (Phys 206b) that you can say that the potential infinite is actual in a sense, namely, in the sense that you can say 'There are Olympic Games' even if the Olympic Games are not all going on at the time, or even if none of them are. This is why, in fact, he doesn't have a problem with the eternity of the world -- it's 'actually' infinite, in the sense of always going on, but it is not an actual infinite, in the sense that every moment is 'constructed' so as to be an actually complete infinite.
