To simplify it, as generally compares/precedes actions/clauses and 'like' compares/precedes nouns.
Like his mother, he frets over the house being clean.
As his mother does, he cleans the house regularly.
However, please also note that 'as' can be used to refer to the function of something/someone: The bigger boy was used as a shield to protect the smaller boy from the bullies.
If you see a noun after the 'as' do not immediately eliminate it; make sure you understand whether it is used to compare or to show the function of something/someone.
Economist GMAT Tutor