Bibinur, can I give you a tip for doing this kind of exercise? These exercises often test both meaning and structure. Look for prepositions - words like 'to' 'from' 'at' and 'with' after or before the gap. Very often the alternatives do have the same meaning. You are quite right in thinking that both options mean 'got better'. But the difference - as Randy has pointed out - is that you 'get over an illness' but you 'recover FROM an illness.' There is no 'from' after the gap, so the only possible answer is 'got over'.

Say "got over". The word "recover" has the form "recover from".

What do you think yourself? What are your considerations?