A newsagent's is typically a small corner shop that sells newspapers. Such shops inevetably sell a great number of other things, cigarettes, chocolate bars, milk, magazines, and a wide variety of small items that one might want from time to time but not want to make a special trip to a large store for. The newspapers are used as a 'loss leader' which entices people into the shop and buy other things once they are there.
A stationer's is a specialist shop selling specifically stationary items, paper, pens, ink, etc.
There is considerable overlap between the two types of shop. Both typically sell greetings cards. Stationer's such as WH Smith's usually sell newspapers as well.
The fundamental difference is that stationer's are almost always part of large chains of shops while newsagent's are almost always small family-run operations.
Hope that helps.