The two answers mean very similar things. 'Looking into' implies an investigation over a period of time. It could also mean a wider investigation of sick leave in general over the whole company. 'Checking' could mean that or could mean that specific instances of sick leave are being checked to make sure that person was actually sick with a valid reason. 'Checking' is probably slightly more hostile, less neutral than 'looking into', at least in British English.

The 2 answers to the question both mean the same thing. However, it sounds a little odd if you were the person who was sick to reply that you were glad they were looking into it. Maybe a more natural thing to say would be, I understand that this is something you need to look into and I will be happy to give you any information you may require.