I don't find your sentences offensive but I don't find them correct either.
First sentence:
The laymen in Europe can't say the difference The Council of Europe
and The European Council.
First of all, this is missing the word 'between', and we normally refer to 'telling the difference', not 'saying the difference'. Also, convention is to use the singular when referring to a generic person like this, as in 'The common man' or 'The man on the street'. So I would rewrite this as:
The layman in Europe can't tell the difference between The Council of Europe
and The European Council.
As Maulik says, the word 'layman' means someone who lacks expert knowledge in a particular subject. However, if there's no obvious area of expertise being referred to (such as medicine in your first example) I would assume that 'layman' is being used in its original sense of a non-ordained member of the Catholic church. So this sentence needs some context to explain what kind of expert you mean (unless you really are talking about Catholics).
Students of European politics know the function of all the different
councils, but the layman can't tell the difference between The Council
of Europe and The European Council.
Second sentence:
This new tax policy mostly effects a layman's life in bad way, not
nobles.
I believe you want 'affects' here (a very common mistake from native speakers as well). I would say 'negatively affects' rather than 'affects in a bad way', to make it easier to tidy up the final clause and use the singular rather than plural for both people.
This new tax policy mostly negatively affects a layman's life, not a noble's.
I wouldn't use the word 'layman' here though, because it doesn't just mean an ordinary person. Being a noble doesn't mean having an area of expertise that others don't. The appropriate word here is 'commoner'
This new tax policy mostly negatively affects a commoner's life, not a
noble's.