There can be overlap between the two phrases; you can sometimes use either one for the same purpose.
Usually they do not overlap.
Think of is often used as a replacement for "remember."
for can be trying to remember something. Examples:
"I am trying to think of the phone number for that store." = "I am trying to remember the phone number for that store."
"I am trying trying to think of my password." = "I am trying trying to remember my password."
"I am trying to think of the title of that book." = "I am trying to remember the title of that book."
In each example, "think of" is being used instead of "remember," but they have the same meaning. This is often used as verbal slang, and is technically incorrect English. (If you write it in your homework, your teacher will mark it wrong. If you say it to your teacher, they probably won't notice it.) In America, you will hear this type of slang every day.
Think of can also be used to imagine something or produce a new answer. It can be used in the place of "create" or even "choose." Examples:
"I am trying to think of a new password." = "I am trying to create a new password." = ""I am trying to choose a new password."
"I am trying to think of a name for my new baby." = "I am trying to choose a name for my new baby."
Think of can also be used to describe finding a solution. Examples:
"I am trying to think of a way out of this mess." = "I am trying to find a solution that will get me out of this mess."
"I am trying to think of the answer to this math problem." = "I am trying to find the solution to this math problem."
Again, all of the examples above would technically be considered slang, but they are constantly used in common, everyday verbal conversations. They are occasionally used in writing as well.
To think about something means that your brain is, well, thinking about. It can be used in many more ways than "think of" can, and is often used to replace "consider," "remember," and many other words.
You can say that you think about anything that is your brain invents, remembers, considers, imagines, "thinks of," solves, and more.
Examples:
"I am thinking about the time that I went to the store." = "I am remembering the time that I went to the store."
"I am thinking about whether or not to marry her." = "I am considering whether or not to marry her."
"I am thinking about what the answer to this math problem must be." = "I am thinking of the what the answer to this math problem must be." = ""I am trying to find the solution to this math problem."
"I am thinking about a purple elephant that sings and stands on it's head." = "I am imagining a purple elephant that sings and stands on it's head."
I hope that these examples help.