There are loads of pubs in Southampton and I've not been to all of them, on this page I summarise the ones I have been to in Portswood.
The Mitre is a fairly modern styled pub, the northern-most pub on Portswood highstreet and tends to draw in crowds during football matches and in the evenings. Due to its location the pub is a popular stop for students who are on a pubcrawl heading South from the University of Southampton. I've never been there when it has been very quiet, so normally have to stand up near the bar and consequently don't end up staying too long. I'm not a big fan of the Mitre, but there's nothing wrong with it, especially if you watch football. I think they also have Sky Sports with 3D TVs, but I've not been there since the addition of these TVs.
There are a couple of pool tables near the front of the pub, but they can get busy when the pub's full. There's a small outside area at the back of the Mitre, with picnic benches, so if you're a smoker you don't need to go outside the front of the pub and they have TVs facing outside next to the windows at the back of the inside of the pub, so if you're watching the football you don't need to rush your cigarette so that you don't miss the game.
In summary, The Mitre is more of a sporty pub, so if you don't like football then don't go there when a big match is on or you'll have to put up with drunken football fans.
Varsity is part of a chain of pubs (like wetherspoons), so it feels less unique than self-owned pubs like the crown, but it's still a good pub. I've been there on pub crawls and for a meal and was not disappointed either time. It's a large pub, so even at busy times there's space to at least stand without being pushed past by others trying to get to the bar. During the day it's very quiet and good for a quiet drink or meal, there's no loud music or loud TV (at least not when I've been there).
There is no beer garden at Varsity, so you'll have to stand outside the front of the pub if you need to smoke and you'll be on the pavement of a high street, but the pavement area is quite wide because the pub is set further back than the shops beside it, so you won't have people walking into you who are trying to get past (unless they're just rude!)
Varsity can get quite busy on Friday/Saturday evenings, because it's in the middle of the main pubcrawl route from the university down Portswood and towards the city centre. Don't let this put you off though, because there's plenty of room at the back of the pub and people don't walk past; thee entrance to the toilets is at the front of the pub and there's no beer garden at the back.
In summary, Varsity is a good pub for eating and drinking, but has no beer garden, so on a hot day, or if you're a smoker, you might prefer to go elsewhere. The pub is in a great location if you're going on a pubcrawl through Portswood and it's big enough to accommodate several pubcrawl groups and still have room for people to sit down and eat.
The Gordon Arms is just South of Varsity in Portswood, it's a traditional style pub and is fairly small, but has a beer garden out the back of the pub, which has picnic benches so it's suitable for people who are drinking that don't smoke, not just smokers. I particularly like that if you're going to the toilets, you have to open a door which looks like part of the bookshelf to get there! It's a nice touch, which threw me the first time I used them.
The Gordon Arms is more of an older person's pub than Varsity and the Mitre, but it's still popular with students who are on a pubcrawl, but you can't seem to escape them wherever you go! I don't think the Gordon Arms sells hot food, so if you're looking for a meal, then you'll have to try somewhere else.
In summary, The Gordon Arms is a traditional pub which is good for a quiet drink and has space outside with seating, if you want to have a drink outside or smoke. If you're looking for a hot meal, then this isn't the pub for you, but if you just want a drink then it's a good place to start. If you're looking to avoid students, then go during the day and especially avoid it on Friday/Saturday nights.
The Richmond Inn is opposite The Gordon Arms and it's similar in style too, so I'm not sure why they decided to put two very similar pubs in such close proximity. The Richmond Inn has two bar areas, with the bar set in the middle. There's a small outside area at the back of the pub, but I haven't spent much time out there, I think it's aimed at smokers more than people looking to sit down and share a drink.
The Richmond Inn tends to be slightly quieter than the Gordon Arms, so if it's too busy in the Gordon Arms then this is an obvious choice. Although the location is ideal for pubcrawls, they tend to stop at the Gordon Arms and skip the Richmond Inn, I'm not sure why.
Like the Gordon Arms, the Richmond Inn doesn't serve hot food, your best bet in Portswood (other than takeaways or supermarkets) is Varsity. The clientele of the Richmond Inn is, again, similar to the Gordon Arms, older people tend to be found there and students aren't often there.
In summary, the Richmond Inn is very similar to The Gordon Arms and is less popular with students who are on a pubcrawl. The pub doesn't serve hot food, so it's not an option if you're looking for somewhere to have a hot meal.