An unofficial answer from someone who's used both services occassionally, but neither regularly. (However, I know there are controllers who work both services who read these forums and will give a more definitive answer!)
London Information: anything at all that doesn't require radar. Things I've used them for have included: activating a flight plan when my departure airfield has been unattended, information on nearby radar facilities, or informing them that I'm entering/leaving UK airspace. They are also, so I hear, very good at giving out information like frequencies, opening times, fuel availability at airfields if you find yourself needing to divert. Although they don't have radar, they will be able to inform you if the airspace you're flying in is known to be particularly busy. They are also listed as a contact frequency for a DACS for some danger areas (I believe they liase with London Military via a land-line to negotiate the crossing for you).
Thames Radar: apart from clearance to cross their zone, they also provide a very good FIS anywhere to the east or south-east of London. In my experience, the service they provide is generally an unofficial radar service, where they will give you a discreet squawk and also give you radar-derived traffic information. They aren't required to do any of this, and the fact that they do is definitely appreciated, especially since they have good radar coverage of one or two bottleneck areas!
FFF
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