###Nominal uses

There are some interesting difference in usage that help illuminate the differences in meaning. I'm using GloWbE because it contains samples from different varieties of English. Here's a corpus search for price¹. Here's one for cost.²

• ... all these tips might help me get a better price.

• Nope today's low gas price is Big Oil's long term plan to shut down its competition
• This is their full price, every other rate is a discount off of this rate.
• ... price rises and falls will not translate into electricity price cuts or increases for consumers ...

This is a context in which you would very clearly not use cost.

Compare the nominal uses of cost:

• I think that libraries will close because of the cost savings in eliminating the employees.

• It does not require isotope separation, a big cost saving.
• Regardless of the personal cost to him or his vested interests

You could probably substitute price in these examples. But it would change the meaning. My observation is that there is some agency that is lost when you use cost in place of price - you pay a price for something, but you incur a cost because of something.

Although it is common, for instance, in order orders, that a party is ordered to pay another's costs, this is entirely consistent with my observation - the costs themselves were incurred because of the legal proceedings, but someone else is paying (the price).

It is at least, unusual, if not ungrammatical to say either:

• I am paying the cost for your mistake.
• I incurred the price(s?) for the professional services.

###Verbal uses

I thought there would be a huge difference in verbal uses, because the definitions are so different³:

price
12. to fix or establish the price of
13. to ascertain or discover the price of

cost
7. (transitive) to be obtained or obtainable in exchange for (money or something equivalent); be priced at ⇒ "the ride cost one pound"
8. to cause or require the expenditure, loss, or sacrifice (of) ⇒ "the accident cost him dearly"
9. to estimate the cost of (a product, process, etc) for the purposes of pricing, budgeting, control, etc

The corpus supports my hypothesis. In purely verbal uses, the overlap exists entirely in that the verbal sense of price are subsumed by part of the verbal sense (the last one; sense 9) of cost.

However, where the past participle (priced; costed) is part of an adjective, priced dominates:

• a higher-priced mobile phone

• low priced beer
• high priced tickets

In practical usage

What does this mean?

Essentially, people tend to use price when they feel they have agency - the price of a house, the price of a car, the price of lunch.

People tend to use cost when they feel like they don't have agency, and the expenses and costs are being placed upon them - the cost of moving, the cost of a free lunch.

This is why, as others have alluded to and intuited, the costs often seem to include non-monetary losses. For instance, to use the moving house example:

• The costs include:
  • The price of a moving truck
  • The price of a conveyancer
  • The price of new furniture
  • The price of setting up utilities

However, you could also say:

• The price comprised of:
  • The price of a moving truck
  • The price of a conveyancer
  • The price of new furniture
  • The price of setting up utilities

In answer to your question

Generally, only the price of something - that is, what you will pay will be listed on a price tag (hence the naming). So, when you ask what how much you will pay, the amount indicated on the tag will be the price, not the cost.

This is a marketing strategy, as well as matter of practicality. If you purchase a car, you would need to spend a certain amount for it, the price. However, in using it, you will spend money on consumables (fuel, tires) and maintenance (servicing). This really can't be listed for any item with great accuracy - it'll depend on usage.