Hmm. There really isn't much of a difference.
I'm disagreeing with TLC here. "To price" can also be a verb, and "cost" can also be a noun.
"What did they price it at?"
"What is the cost?"
They mean essentially the same thing. One small difference I can think of is that "cost" can be an abstract concept, while "price" typically refers to numbers, but that's not a strict rule.
The other small difference is that "price" is something attached more to the object, while "cost" is attached more to the subject. This one is probably the only good way to differentiate which way to use them in a sentence.
"How much does that cost (me)?"
"What price (of the object) are you asking for?"
"The cost (incurred upon a country) of freedom is quite high."
"Did they price (the object) too high?"
"Courtesy costs (a courteous person) nothing."
"He has a price on his head."
The parenthetical sections are all implied in those sentences and wouldn't actually be written. There are some times where the words can be used interchangeably, but the focus of the words remains. For example, to rewrite a previous example: "What is the price/cost of freedom?" Both of these sentences can work. However, "What is the cost of freedom?" is asking about what it will cost a PERSON to be free, and "What is the price of freedom?" is asking how much freedom itself is "priced".
But once again, despite their differing focus on either the object or the subject, they should be looked at as synonyms.