Glen Eyre, Wessex Lane and Mayflower are generally considered to be the three main halls at Southampton. I know you didn't mention Wessex Lane but I'll give some brief info about all three just in case.
Glen Eyre
Location: 5-10 minute walk from Highfield Campus
Glen Eyre is the closest of the three main halls to Highfield Campus, being a 5-10 minute journey just up the road. There is a range of hall sizes, typically about six to a kitchen though some spaces have 10 or more. Slightly less of a party atmosphere than Wessex Lane but still very sociable. Despite Mayflower being more modern, the facilities are more than enough at Glen Eyre, with a fair amount of washing machines (though for all halls in general I recommend doing your washing on Mondays while people are still out in lectures) and very good kitchens. There's a small bar within the halls complex (though not close to any halls blocks, so noise isn't an issue), plus lots of green spaces and barbeque pits for when the weather isn't terrible. Big Sainsburys is a five minute bus ride away in Portswood, good for your weekly shop, and if you want to stock up on little bits like milk or bog roll during the week there's a Co-op, a Sainsburys Local and a Tesco Express, all around a 10 minute walk away. Being close to Highfield means you're close to a lot of events during Fresher's that take place in the Student Union building, as well as The Stile and Stag's pubs. The main appeal of these halls I think is the location, as well as the pretty comfortable living spaces.
Mayflower
Location: 15-20 minute bus ride from Highfield Campus
Like you said, Mayflower is more modern. While it's the furthest away from campus of the three main hall complexes, it has the best facilities. They've recently spent a lot of money installing a new gym inside the halls complex itself, meaning you don't have to endure a sweaty bus ride back from Highfield gym. There are city centre buses (coded U1C) that go pretty close to Mayflower Halls, but there's also a direct bus route to and from Mayflower, so despite the longer journey, accessibility isn't a major issue. Obviously, being in the city itself, you're close to a lot of shops, including a big Asda, and there's also a Co-op actually in the halls complex for bits and bobs. There's Mayflower Theatre directly next to the halls, which gives cheaper tickets to students (entrance to Sweeny Todd was £5, last I heard). You're also pretty close to the clubs in the city centre, where a number of Fresher's Week events take place, and of course, you're the closest halls to Sprinkles, the best ice cream bar in town. While you may need to get up earlier for lectures, the main appeal of these halls is its close proximity to the city centre and its excellent facilities.
Wessex Lane
Location: 5 minute bus ride from Highfield Campus
Wessex Lane halls are pretty big, with a lot of first year students ending up there. The rooms are slightly less good in terms of quality, but it's still a decent living space. It offers cheap accommodation, too, and you have a much greater chance of getting in these halls if you put it as your number one choice compared to the others. There are two bus links that lead from Highfield to Wessex Lane halls: the U1W, which is dedicated specifically to Wessex Lane, and the U1A, which goes towards the airport but takes the same route as the U1W. The bus rides are frequent and very short, and go past all the small shops I mentioned in the Glen Eyre paragraph. There seems to be a bit more of a party atmosphere at Wessex Lane, though that may just be because there are more people. If you're interested in the archery, rifle or rowing societies, they're based a few minutes walk away from the halls. The facilities are alright, and you obviously have the essentials like washrooms and reception. Really, Wessex Lane is nothing special, but it's still a good place to live. The main appeal of these halls is the number of rooms available, the cheap pricing of the rooms, and the relative closeness to Highfield Campus.
With all these halls, bear in mind that there are different types of accommodation within the halls. I'm not sure about Mayflower but Glen Eyre and Wessex Lane may offer you halls of around 20 people, or they may offer you halls of six. The allocation of halls is a bit confusing as well, as there are different sections within the halls. In Glen Eyre, you have the main Glen Eyre halls, but you could also be put in Chamberlain, which still counts as being part of Glen Eyre but contains the Hartley Grove and South Hill halls (don't worry, though, as if you look on Google Maps you'll actually see that these halls are slightly closer to campus than main Glen). Wessex Lane contains Montefiore (or Monte) and Connaught halls, the latter of which I believe to be catered halls. Just something to keep an eye out for.