I differentiate by outcome in Literacy. All children do the same task- but they'd have their own success criteria. I would never dream of giving a 'fill in the gaps' activity to a child.
This would look like a child who is still developing writing ' I went to the shops' and a more able writer writing 'First, I went to the shiny new shops to buy some yummy chocolate'. The developing child would then be encouraged to add adjectives as part of their next steps.
With Maths I teach for mastery and we all solve the same problems but with different resources. Sometimes the number work is different, in that some children won't be able to bridge 10 or work with larger numbers- so activity sheets will reflect that. As a result of this, I do use ability tables. This means support can be targeted easily and helps with classroom set up. All children have equal access to the challenge box when they are done.
Mixed ability work is common though and tables are fluid. If I thought a 'middle ability' child was ready for extension as a result of my questioning- then I would give them the 'higher' set of questions and vice versa, if a usually bright-at-everything child found some concepts difficult.
It would take me an extra 30 minutes each lesson that I don't have to wander round and place different activities and resources individual to each child's seat. But there is so much benefit too- teachers can cluster the students by similar goals. One of my 'ability' tables is due to 'handwriting'. This means I can place myself or a TA there to praise and encourage legible writing... or lots of play dough for the session starter for them to do fine motor skill practice.
Schools are 'cagey' about it because I have had absolute hours of my time wasted by parents demanding to know why their child is on Blue table when they should be on Orange. 
In my opinion- ability tables are the best way to differentiate, as long as it is complimented by lots of mixed ability work and the groups are fluid and changed often.
Some schools do chilli challenges where children self differentiate ('mild, medium, spicy') and I'd be wary of any schools that do that. It looks all 'promoting independence' but the brighter children end up choosing work that doesn't push them and those who need support can be attracted to 'big numbers'. Requires a lot of running around from the teacher to be done properly, which stops it being self differentiation - if they notice the correct work isn't being done...