Firstly, note that not all jobs in the UK with the title "Lecturer" will be at universities. Many further education colleges (which mostly teach teenagers, between 16-18) use the job title lecturer, as do several private training companies. I'm pretty confident that no one in the UK works as a university lecture and is only paid £34k. But there could be another reason for the difference:
UK academia pays people on a single pay spine, nationwide (except, oddly, for Imperial College). This is divided into approximate 50 pay points (I think). Universities may decide which points on that pay spine relate to which jobs. That said, there isn't that much in the variation between universities of a similar class (except Imperial).
The job title "Assistant Professor" is new to the UK, although an increasing number of universities are using it (Including Oxford, Warwick, Leeds etc). At the universities that have changed over, it was a direct substitution for the title Lecturer (everyone who was Lecturer became assistant professor).
There are two be categories on university in the UK, the pre-92s, and the post-92s. The post-92s are so called because before 1992 they were not called universities, but rather "polytechnic colleges", and taught vocational subjects, while universities stuck to academic subjects. The post-92s tend to be more focused on education that on research, tend to have more students, and importantly for this discussion, pay their staff less.
For example, there are currently jobs available for the lecturer grade at Manchester Metropolitan University (post-92) starting at £37k (spine point 30), Huddersfield (£39k), Birmingham Newman (£37k), while at The university of Sheffield Lecturer jobs start at £46k (spine point 38), £44k at York University, £45k at University of Leeds Leeds
The majority of the universities that use the Assistant Professor title are older, more prestigious, research intensive institutions, but not all such institutions use the title Assistant Professor.
So, the real distinction here is not that Assistant Professors are paid more the Lecturers, but that entry level faculty are paid more at pre-92s (some of which use Assistant Professor, and some Lecturer) than at post-92s (almost all of which use Lecturer).
