OK, I went to Leicester and my girlfriend throughout uni was at Birmingham, I now do the LPC in Birmingham.
Firstly both unis have quite a similar feel, they are campuses on the edge of a city centre but in which students largely stay confined to the bubble of uni (either campus, Selly and the Vale or campus, Evington/Clarendon and Oadby).
Brum's halls are nearer to campus, but the bus is no hassle and I think Leicester's halls were marginally better, my ex agreed so I don't think it's just me being biased, the sense of community at Oadby is incredible, however the Vale is still a lot of fun.
For obvious reasons Leicester has a friendlier more intimate feel, it's half the size. However the size means Birmingham can offer a huge range of extra curriculars and has lots of options for things like study abroad.
Careers wise the fact Brum is RG is largely irrelevant, I have a hunch Leicester may join soon itself anyway and if not you'll find most recruiters aren't 100% clear on the RG's make up anyway, they just use it as shorthand for their own definition of 'good uni', which Leicester falls into. Where Brum has an advantage is the sheer number of firms who go to campus, as there are simply a lot more straight A students they can hit there, by contrast Leicester attracts fewer firms but it does mean you can be certain those firms are keen on Leicester and like their graduates, rather than just playing the numbers game. Finally most Birmingham firms recruit hard at both unis, but there are slightly more workshops and the like at Brum and a lot of Birmingham grads on my LPC are going to Wragges, Gateley, DLA Birmingham etc. If you do wish to stay in the city after graduation you'll find this is a lot more common for Birmingham grads, I think there are more Leicester grads in Birmingham than Leicester, and it is nice to have that ready made friendship group still there for when you start work.
Both cities have options for the LPC/BPTC if you wanted to stay around to do it, and if you graduate after 3 years you probably will as loads of your mates will be doing 4th years.
Extra curricular wise, Birmingham has more options but I found participation per capita seemed noticeably higher at Leicester so I think that's pretty even. Sport wise if you're really good Birmingham's where it's at, if you're just alright and want a chance to pull on uni kit of a Weds afternoon Leicester's the better bet.
Both have quite nice going out areas in the city centre for you to explore a bit in 3rd year for a break from the student seen, be that the Mailbox and Colmore in Birmingham or the are around Hakamu in Leicester. Both also have plenty of cultural events, from the Phoenix (Leicester) and Electric (Birmingham) to the Golden Mile and Balti Triangle.
Night out wise Leicester is more Union focussed (twice a week) with most Brum nights out happening on Broad Street, but then Birmingham students go mental for Fab at the Guild on a Saturday and it's the main night out you'll go to in second and third anyway. There's a greater range of clubs in Birmingham obviously, but Leicester has a student night out for every night of the week still.
HTH a bit, let me know if you have any qs (particularly about Leicester but Brum as well to an extent).