Each area is covered by three 'multiplexes' (which are simply bundles of channels). Those multiplexes might be transmitted from different locations, resulting in differences in signal strengths.
One multiplex transmits all of the BBC national stations (but not BBC local radio). It's clear that both of your radios are receiving that multiplex.
A second multiplex transmits national commercial stations . Digital One owns that multiplex, so their own stations (Classic FM, Absolute Radio & TalkSport) are included in the package. They also broadcast content from other national broadcasters, including Planet Rock, Amazing Radio & Smooth Radio. Once again, it seems that both of your radios can receive that multiplex.
The third multiplex carries local radio stations (both BBC and commercial). It seems that the signal from that multiplex is weaker than that of the other two multiplexes. Your Sony radio clearly has a poorer 'sensitivity' (i.e. its ability to handle weak signals) than the personal DAB radio which you also use.
Start by using the postcode finder, here, to see whether you're 'officially' in the reception area for a 'local' multiplex:
http://www.getdigitalradio.com/
If local stations are listed on the results page, your Sony set isn't really up to the job and you should consider insisting that the retailer exchanges it for a different model. However, if local stations aren't listed, you can only consider yourself lucky to have the small radio with exceptional sensitivity which can bring in stations which are still officially beyond your reach. (Many parts of the country still can't receive a 'local' multiplex. I can't, here in Suffolk. It will be several years before all of the planned 'local' multiplexes are on air).
Chris