So, in the text we repeatedly find this idea of something being 'in between' two opposite extremes. Love of wisdom is 'in between' wisdom itself and ignorance (Symp 203e–204b); eros is neither a god nor a mortal, but 'in between' (202d); and he is also not ugly nor beautiful, but again he's 'in between' (201e–202b). The whole setup is of course designed to explain the drive that our love (for beauty, or for wisdom) so obviously has: by attributing some characteristics to eros as a god (or whatever he is if he's not a god but something 'in between' gods and mortals) we get some clarity about our idea of love. So let's get clear about in-betweenness as a first step.
1) When it is suggested (by Diotima, who is teaching Socrates here) that eros is in fact a philosopher, the full text reads:
He is in between wisdom and ignorance [...] In fact, you see, none of the gods loves wisdom or wants to become wise — for they are wise — and no one else who is wise already loves wisdom; on the other hand, no one who is ignorant will love wisdom either or want to become wise. For what's especially difficult about being ignorant is that you are content with yourself, even though you're neither beautiful nor good nor intelligent. If you don't think you need anything, of course you won't want what you don't think you need. (204a)What does it mean to be 'in between' wisdom and ignorance? Sometimes being 'in between' can mean to take a position on some spectrum, being neither at one nor the other extreme, but somewhere in the middle. For instance, water (under normal circumstances such as pressure and so on) will have some temperature between the freezing point and the boiling point: a given bit of water will usually be 'in between' these two extremes.
But that seems not to be a good model for what we're talking about here: it's not as if the amount of wisdom in a given person were somewhere on a spectrum between zero (total ignorance) and some maximum value (full wisdom). Not only is wisdom surely not the sort of thing that can be quantified in this manner, but this model also doesn't include the element of wanting that seems to play a role: those who are ignorant aren't just ignorant (at zero position), they also don't want any wisdom, and likewise, those who are wise are not just at the saturation point (or maximum position), but they also don't want any wisdom. So there is another dimension here that we have to consider.
Let's try and plot this on a diagram. There are the two dimensions of in fact having wisdom and wanting it, and so we can find four principal constellations:
First, there are the gods who both have and don't want wisdom (on the upper left), and second the ignorant who don't have but also don't want wisdom (on the lower left).
Now, third, we also have the lovers of wisdom, who are neither wise nor ignorant, and those occupy the lower right.
There is a fourth position in this diagram, and we (and Plato) must ask ourselves whether it would be a possible constellation, and if so, what about those who would occupy it. Could someone be both already wise and still wanting to be wise?
This is an interesting question, and it has an interesting answer: in general, Plato thinks that, yes, this is a possible constellation, and those who already have something can at the same time desire it — but it is a desire not to get that thing, but a desire to keep it in the future. Socrates discusses this earlier on (in his interchange with Agathon), and he brings examples such as "maybe a strong man could want to be strong [...], or a fast one fast, or a healthy one healthy: in cases like these, you might think that people really do want things they already are and do want to have qualities they already have". In these cases, Socrates argues, "what you want is to possess these things in time to come, since in the present, whether you want to or not, you have them." (200b–d)
But even though in general there seems to be such a position as the upper right in the diagram, in the specific case of wisdom it's not acknowledged: none of the gods loves wisdom. What makes the case of wisdom special, I suspect, is that in this case who is already wise are the gods, and they are immortal and presumably will remain wise indefinitely, for all time anyway, so there is no point in attributing them a desire for continued wisdom. (Still, I think, even if this conjecture is correct, there would be the theoretical possibility of a mortal becoming wise, in which case that mortal could have the valid desire to remain so in the future, and then that wise person might have a love for wisdom even in being wise already. Since this is denied explicitly in the text, my guess is probably false, but then I don't see what makes the case of wisdom special compared to those other cases Socrates lists.)
I think this understanding of in-betweenness in the philosophy (love of wisdom) case is a good start. Let's check next how it fares with the other examples in the text.
 I'm using the Hackett edition by Alexander Nehamas and Paul Woodruff, Indianapolis: Hackett 1989.