A good interview question!
Broadly speaking, attainment is standards and achievement is progress.
Attainment is summarised in test scores and examination grades.
So, if a child is expected to get 5 GCSEs and they get 7 they have got (attained) 7 GCSEs and made good progress (achievement).
It may be better explained when achievement goes the other way.
A certain school had good GCSE results.
The children were leaving at (say) 4 or 5 or 6 points above the national average.
The staff and parents were happy (especially the parents!).
On inspection, it was pointed out that the children were entering the school at 7, 8 and 9 points above the national average.
So, the achievement (progress) was negative/backwards (appalling).
The school was placed in ?Special Measures?, many of the staff were in shock, the parents were very puzzled and angry. There was much explaining and hard work to do.