เก่า is simply old. It describes physical appearance of things that looks old. You can sometimes use it with people if you refer to positions (boyfriend/girlfriend แฟนเก่า, manager หัวหน้าเก่า, owner เจ้าของเก่า, etc) เดิม refers to previous things. คนเดิม = same person (previous person that you knew) รถคันเดิม = the same car (that you used) บ้านเดิม = the same house (that you lived)

เก่า is equivalent to 'old' in English. (ex: หนังสือเก่า) เดิม is sort difficult to explain. Its meaning differs in different situations. It can mean 'traditional' (เพลงไทยเดิม = Traditional Thai song) or it could be something like 'in the past' เดิมแล้วเขาไม่ใช่คนอย่างนี้ = He wasn't this kind of person [in the past] or it could mean 'same' คนเดิม = same person ฉันใส่เสื้อตัวเดิม = I wear the same shirt :)

Both shared the same meaning, "old." But they were used for different purposes. "เก่า" will be used when ... 1. you describe how old something is (but you can' say "He is so เก่า," because that is so wrong.). You can only describe a subject, for example: ของเก่า (which means antiques). "เดิม" will be used when ... 1. you want to describe the oldness of someone or something, in a sense of how familiar it is to you. I can make an example as shown: เจอกันที่เดิม means Let's meet at the same old place (it implies that you and your friends have met at that place before.). Or it could be ไปที่ไหนก็เจอแต่คนหน้าเดิมๆ means Everywhere I go, I only see the old faces (it implies that there are nothing new. It was as usual. No one new. No excitement in life. Bored.) Hope those explanation helps. :)