"Definitely" in practice is more certain than "certainly". "Certainly" is more often used when granting a request. Think of "definitely" as "This is true or will be with no doubt whatsoever," and "certainly" is more like "This is true or will be." An example would be, "Do you think you will win the race?" You could answer "Yes, definitely". This would mean "There is no doubt in my mind, and I am completely determined to win." If you answer "Yes, certainly," you would be saying "Yes, I will win." If someone asked "Will you please go to the store?" You could say "Yes, certainly". You are saying "Yes, I will." If you answered "Yes, definitely," then you are saying "Yes, I absolutely will." This would imply you are not simply granting the request, but that you have an additional good reason to believe that it will happen. (Perhaps you have forgotten to go to the store previously, and you are determined that this time you will remember to go.)

'definitely' is used to emphasize that you mean 'Yes' 'certainly' is used to emphasize that something is true or will definitely happen. 'Certainly' is also used to give permission or to express agreement. So they are answers to different kind of questions. When someone asks you for example: "Are you going to call her?" You will rather use: "Yes, definitely" To express that you are going to call without doubt and emphasize the 'yes'. When someone asks you: "Do you think you have enough cash to invest in this project?" To express your agreement ,you will rather use: "Yes, certainly" When someone asks: "Do you mind if I used your cellphone ?" To permit the person to use it you will answer : "Yes, certainly" Notice that you could use both many times interchangeably,but this is the linguistically accurate difference to be taken into account, when choosing between both.

No difference.