Lancaster is higher up in the overall uni tables, top 10 in all of them.
That depends which tables you're looking at. The international QS rankings put B'ham at 14 in the UK(79th in the world) whereas Lancaster comes in at 20(131)
www.topuniversities.com/university-rankings-articles/world-university-rankings/top-universities-uk-2019
As to comparing the two unis... I went to one (40 odd years ago) and live quite close to the other.
Birmingham has one of the best campuses going, IMO (they're currently doing some redevelopment, so not sure how it looks right now), and the main halls site is really nice, a short walk away around a lake. There's Selly Oak on one side (curry houses), leafy Edgbaston on the other, and a variety of suburbs accessible for rentals. The local train line gives good access out to some of those, but also into the city centre. You've got the advantages of being able to get into a big city but also the advantages of a proper campus (much nicer IMO than somewhere like Manchester).
Lancaster, OTOH, as PP have said, is a campus about 3 miles outside a compact, pleasant but quite quiet town. Regular buses, some good cycle paths. If you're an outdoorsy sort of person then you can get out into to proper countryside (which is a lot harder from Brum!). Apart from the halls on the uni site (which I assume are mostly for first years?), there seems to be a lot of student lets in the town - there seem to be some large student accommodation blocks going up at the moment.
So, depends on what sort of environment the student likes, but obviously also on the subject - some of the Lancaster departments are rather small, I think.