"Başka" is usually being used when you do not know the object you point to. For example if there are a lot of cars in a car gallery and you say " I do not want to buy the first one, I want to see another car = Ben ilk arabayı almak istemiyorum, başka bir araba görmek istiyorum." "Diğer" is usually being used when you do know the object you point to. For example if there exist two cars which you can see them and you want to say "I did not like the first one, I would like to buy other one = Ben birinci arabayı beğenmedim, diğerini almak istiyorum."

You can definitely use them interchangably. But I'll shoot some examples here. Başka bir deyişle, bu böyle olmaz. Diğer is less preferred. Diğerini bana uzatır mısın? Diğer means 'the other one'. Başka bir tane uzatabilir misin? As you see başka is more like 'just another one'. Diğer is clearly stated or decided. So başka is like just another one. Diğer seems like the other one. Hope it helps.

diger is used if there are only two special things.