Electron affinity for Cl is not 'greater' as such. The fact is that the electron affinity of F is smaller. This is because of F's small size and electron density. This makes it unfavourble to add another electron. Although it does become stable after it gains an electron, there is some repulsion that extra electron undergoes.
As far as same chemical properties of elements in same period is concerned : electronic configuration. I strongly suggest you look this up and learn what it actually is and how it works