Although (as part of Spain), the Canary Islands are within the EU, they've always had special tax status, which means that they're regarded as 'non-EU' as far as personal imports are concerned.
So, just as when travelling back from other non-EU countries (such as the USA), you're restricted to bringing back the following without getting clobbered by HMRC:
16 litres of beer
4 litres of non-sparkling wine (excluding fortified wines, such as port or sherry)
EITHER 1 litre of spirits OR 2 litres of sparkling or fortified wines.
200 cigarettes OR 100 cigarillos OR 50 cigars OR 250g of tobacco.
Those rules won't change after the UK leaves the EU. However (unless there's some type of deal struck) they will then apply to imports from ALL EU countries (e.g. France or Spain) as well as to the Canaries and non-EU countries.
At present UK citizens travelling to the Canary Islands are entitled to receive the same health care as local residents get, simply by presenting their EHIC. If there's a 'no deal' Brexit (or if there's a deal that doesn't include health care), all EHICs held by UK citizens will cease to be valid.
At present, mobile phone companies are obliged to charge you the same rates for using your phone in the Canary Islands as they do in the UK. (i.e. 'roaming charges' are banned). That obligation will cease under a 'no deal' Brexit (or possibly with a deal that doesn't cover roaming charges), so you could end paying a great deal more to use you phone abroad.
Nobody can be certain as the effect that Brexit will have upon the value of the pound but the majority opinion is that you'll get fewer euros for each pound when you come to exchange your money. At present the official exchange rate is €1.09 for each pound but it's closer to parity (i.e. £1 = €1) in many places. That's expected to fall a bit (possibly to £1 = €0.90) with a deal for Brexit or rather more (possibly to £1 = €0.80) if there's no deal.