The OP says experience a bit of the culture. Well these days some booming nightclub in Saigon with blokes drinking whiskey whilst being served fruit platters or a multi-storey coffeeshop is as much the culture as pagodas etc.

Anyway I think I'd go with Hanoi if it was one location only. It's just prettier. The downside is that hotels/service is generally a bit crappier - and doesn't really show VN at its best. The OP hasn't said when she intends to travel. Autumn/Winter/Spring in Hanoi can be very nice - you can walk around the streets very comfortably and there are certainly plenty of delights to be had in the old quarter. Saigon tends to be very hot (especially for a Brit) and you end up hopping from one air-conned environment to another. But in the summer, Hanoi is worse than Saigon and they seem stingy with the aircon so it can be rather too much.