For Sony, B series is 2014 models, and C series is 2015 series (which is by now half new.. since it is available since a few months ago and announced even earlier, sometimes around first quarter this year, if I remember correctly).
600 and 800 are 2 different series with different feature set, and different price range - 600 is cheaper. Depending on your usage, the new Android TV platform available on some of the Sony mid-range and flagship series this year, may or may not be a feature that is worth the money, to be honest. You might want to consider your mum's preference and go big with the older model.