For illustrated stamps, you really need not look any further than the Czech Republic. I'm not entirely certain, but it does seem like 99% of their stamps are illustrations rather than photographs.
From their lovely depictions of flowers, like this tulip from 2010
or the cyclamen from 2007, or the azalea from 2008 (all by graphic designer A Khunova)
to the Postcrossing stamp designed by Maria Nogueira in 2015
to perhaps the most famous Czech artist, Alphonse (or Alfons in Czech) Mucha
this is his 'Zodiac', which was originally designed as a calendar in 1896
to this 2014 stamp promoting Good Luck, with a dog chewing on a four leaf clover.
It was designed by Jiri Sliva and I really have no idea why the attached bit on the left is of a dog tied to a lamp while reading a book. But - and I did not know this - if you find a four leaf clover and you are wanting to find a mate (husband) an unmarried woman is supposed to eat it and the first unmarried man she sees afterwards will be her true love.
more countries from around the world have illustrated stamps, find some at Sunday Stamps II