I generally Automate settlers as they are smarter than they used to be in previous Civs and I don't want the hassle of figuring out what to build which they can do for me and 90% of the time they do pretty good.
Things they do well imo.
Build improvements based on specialisation/puppet status of the nearest city i.e. near commerce/puppets cities they'll spam Trade Posts, near production cities they will generally build more mines etc and near growth cities they spam farms and near default cities they will try and balance it out Early on putting mostly farms but later on down the line they will mix in trading posts as well.
Hook up your trade routes pretty well (not completely optimal but not bad).
Hook up Luxuries pretty fast (all though they don't seem to differentiate a luxury you don't have over one you already do nor that Marble is better than other luxuries)
Hook up strategic resources fast. Including those like coal and uranium when the tech required is unlock (as long as you have improvement replacement left ON)
Upgrade roads to Railroads fast.
Things they do poorly...
Run back and forth between cities too much, like someone said before they don't concentrate on a city and move on they often move around too much.
Get scared of barbs too easily wasting you valuable time on the resources.
Run towards waring city states too much to improve land and get captured. And improve land near your enemies borders because they can't "see"/have line of sight on the enemies units therefore think "oh it is ok to improve over there, there's no one over there that can capture me... DOH they were hidden durrrr...."
Prioritise hooking up stone too low. They only seem to value this based on the +1/(2 at chemistry)
from just the quarry rather than the fact it unlocks the stoneworks. Therefore they might take forever to hook this up.
(and for me the automated workers when playing with the Netherlands seemed to be bugged as they NEVER build any food improving improvements except Polders (where they can) even on land that Polders can't be built. So they won't build ANY farms ever nor will they build Pastures on sheep or cows (they do on horses) and won't build Plantations on Bananas (again they will on Luxuries), I'm assuming there a bug in the prioritisation of food based improvements due to Polder being a +3
improvement that causes this)
Overall this is what I suggest if you want to use Automated workers
Use Manual Workers near the beginning so you can hookup the resources and use better judgement then the Worker AI when dealing with Barbs.
Use manual workers to hook up Marble before another available luxury. So you can build those wonders fast. While automated workers will improve marble very fast it will not alway hook it up over another luxury and even if it does it will wait till its finished its task to do so. At the start of the game you want to get that marble ASAP and if you need to interrupt a worker building a farm then you can if they are on manual.
Automate during peacetime/during war with AIs that are well away from where your workers would want to work, you don't have to worry too much during War in general unless its a really close border dispute or a CS allied with the AI is right on your doorstep.
Make workers manual during war time with very close enemy Civs or CS.
Make a Worker manual anytime you want to make sure you get a resource such as stone hooked up so you can build stoneworks or they are seemingly neglecting something like Uranium or Coal when it appears.
Rest of the time just let them do what they want its pretty good in general.
Also I leave replacing improvements turned on as then when you unlock a late game strategic resources they will hook them up and if you puppet a city they will put Trading post over the top of farms and mines. They will also change the land based on any new changes of circumstance such as techs with improvement upgrades, placing trading posts down later in the game on top of farms or if you change the specialisation of a city.