We say "I'll keep that in mind" (no "my") when you want to assure the speaker that you have taken note and value what they say.  You have "put" their information in a certain place in your mind for when you might need it.  You don't normally need to use it now, and that is why it can be a polite form of refusal.

"Do you want to buy a computer from my store?  I'm not looking for a computer now but I'll keep your store in mind if I ever do need another one."

"think about" = consider.  e.g. Have you decided where you want to go on holiday? No, I'm still thinking about it.

If you say, "I'll think about it", in a sales or romantic context, it can mean "I may be interested - tell me more / try harder" but it can also mean "I'm not interested". Context is everything! 

very good questions.

It's "in which situation can I use" when asking a question. 

You can generally put the verb associated with the subject before the subject when asking the question but after the subject when making a statement. 

question: Can I use...?

statement: I can use...

I'll keep that in mind: means to remember to apply it to a detail of decision making.

Example: Sara told me to keep in mind that there is an extra charge for suitcases over 50 pounds. When I pack my suitcase, I want to remember this so as not to have to pay extra.

I'll think about it: means to consider a possible different action than the current action. 

Example: Sara invited me to join a Zumba class with her. Right now I only exercise a little so I said that I will think about it. 

I have heard these used interchangeably and am not completely sure whether they are exactly synonomous. 

I'm interested in others' definitions as well.