Most BrE-AmE dictionaries will tell you that BrE jumper = AmE sweater, but this is a little misleading and far from the whole story. When referring to knit(ted) garments, AmE sweater has much broader application than BrE jumper, which refers only to (generally long-sleeved) pullovers--that is, they are donned by pulling them over the head. In BrE, jumper stands in contrast to cardigan, a word that is used in AmE, but sweater is used frequently in AmE to refer to cardigans as well. So, AmE sweater is a superordinate term or hyper(o)nym, which includes cardigans and pullover sweaters. In BrE, jumper is not the hyperonym of cardigan, but kind of its 'opposite'.
Jumper in AmE is a kind of dress, called a pinafore (dress) in BrE. (Both dialects have the 'apron' sense of pinafore.) In other words, it's a sleeveless dress that's made to be worn over a blouse or other top. Thus my mother, who finds cross-dressing unexpected and hilarious, always has something to say when Better Half says he's going to put on his jumper.
Another sweater that is not a jumper is the (AmE) sweater vest (illustration from this catalog(ue) site). Now, there are two reasons why this isn't called sweater vest in BrE: (1) sweater is AmE (as already established!), and (2) in BrE vest is generally used to refer to (more typically AmE) undershirts (with or without sleeves) or sleeveless undershirt-like things worn by sports players. In AmE, on the other hand, a vest is a sleeveless garment for the upper body that's typically worn over a shirt. This includes the kind that one finds in three-piece suits, which have buttons up the front, and which BrE speakers call a waistcoat. Vest was once used in BrE for what are now called waistcoats--originally the term for a more complicated garment:
The earliest waistcoats, intended to show through the slashings and other openings of the doublet, were often extremely elaborate and costly. They were sometimes provided with sleeves, and appear to have reached to or below the hips. (OED)So, Americans kept an old (but certainly not the original) meaning of vest, while the British adjusted the meaning of another term. A related term that I've only heard in the UK is gillet (also gilet), for a type of furry waistcoat/vest that became fashionable a couple of years ago. (Here's a photo.) I was questioning whether I've only heard it in BrE because it's only been fashionable since I moved here, but most of Google results for gillet + fur are from the UK, so I'm suspecting that it's a far more common term in BrE these days. On an American catalog(ue) site, I find similar items described as fur vests.
But I've got(ten) away from the question: what is the BrE for (AmE) sweater vest? It is, in my confused experience, tank top. Here's a so-label(l)ed photo from a UK retailer. The experience [of hearing of a dean coming to work in a tank top!] was confusing for me because of the AmE meaning of tank top: a sleeveless undershirt (nowadays often worn as an only-shirt). I was wearing one of those today, so had the opportunity to ask Better Half what he'd call such a thing, and his (sorry, honey) rather unsatisfying answer was 't-shirt', later adapted to 'sleeveless t-shirt'. A more precise BrE term is singlet (as one can see here), but it's not a term one hears a lot these days. Such undershirty things are likely to be called vests, as one can see when searching 'vest' on the Next [UK clothing retailer] website.
This brings to mind another (colo[u]rful but unfortunate) Americanism: wife-beater, which is a slang term for the type of tank top/vest that Marlon Brando wore in Streetcar Named Desire. Slangcity.com claims (I've never heard it) that wife-beater is also BrE slang for Stella Artois beer--which brings one back to Brando and Streetcar (Steeelllllaaaaa!).
Getting back to sweater and jumper, there are more ways in which the former is more general than the latter. For example, I have fine-gauge, short-sleeved knit(ted) tops (like this one on Knit Sisters) that I'd only wear on their own--not over another shirt/blouse--and that I'd call sweaters. I'd not feel comfy calling such things jumpers in BrE, though. Searching summer-sweater on Google Images brings up images of both short-sleeved and sleeveless tops and lightweight, long-sleeved sweaters/jumpers, but searching summer-jumper just results in lightweight, long-sleeved jumpers/sweaters and AmE jumpers (the one short-sleeved one is a red herring: it's on the same page as a long-sleeved one that has the 'summer jumper' label). What would one call a 'summer' sweater in BrE? My best guess is that it's just a top. (BrE-speaking 'summer sweater' wearers, what do you think?)
And speaking of top (once I get going, I just can't shut up, can I?), I find that it's used much more often in BrE than in AmE. And in AmE, one is more likely than in BrE to call a woman's blouse or top a shirt. I'm not saying that these terms aren't used in both dialects, but just that their frequency/commonality seems to be different--at least in the forms of BrE and AmE I've been exposed to. But on that intuitional note, I've got to go bed...