Price and worth are similar, but a true price only exists when we are actually selling, whereas we can speculate about worth at any time. Also, if we actually do sell, and prices are negotiable, we will set our price higher than what we think it is worth.
Of course these generalizations about usage have many exceptions. Expressions of worth that are vague (worth a great deal) or dramatic (worth the world) refer to non-monetary value. We aren't anticipating the selling price of our civil rights or our children. A clever economist could probably calculate a cash value of civil rights, perhaps by comparing the social health of a place where they are common with one that lacks them to some degree, but by law and tradition it’s profoundly wrong to assign a price to them. Similarly, when we say we are counting the cost, or ask At what price?, we are not actually expecting a discussion of money.
Economics offers theories (see Story) that explain the observed behavior of buyers and sellers in markets, exchanging money for goods and services. In this real world we see that price, value, etc. are similar, so it is unsurprising that economic theory offers explanations of why they should tend to be the same. Economists have also examined various sorts of mismatch and offered theories to explain them, but equilibrium theories predict that, in the long run, value, price and worth will converge.
Equilibrium theories are still popular with economists who are ideologically opposed to government intervention (see Austrian school and Chicago school), but the major dividing line between Classical economics and modern thought is appreciation of how poorly the premises of equilibrium and rational individual choice describe actual economic behavior. The economy is a Dynamical system that not only never reaches equilibrium, but is also chaotic. Our judgements of value and worth are consistently biased in ways that do not correspond to economic rationality.
This is the essence of Keynes’s Animal Spirits, the mood of the market. The market price is subjective value rendered specific by the market. The market does indeed average individual value judgments, but our behavior deviates from the economic ideal in consistent ways, and is strongly influenced by our social network, so what market converges toward is The Collective Subjective.
Modern economics also recognizes that there are cases (such as environmental pollution) where an unregulated market does not lead to a desirable outcome. This Market failure can be understood as a long-term mismatch between price and value.
Prices vary, even when the theoretical value of the products and number & wants of the potential buyers assigning worth to the goods do not– Similarly, if the demand for something goes up–if it’s worth to the people making up the collective subjective rises–the price should also go up, but not immediately. The disconnect may be small, but it exists. The fact of the lag-time's existence can be described as supply and demand in action. True enough. But that does not change the fact that supply and demand are not static, and therefore can be more broadly described as a demonstration of a process in action, proof of one situation becoming another situation, rather than instantly jolting into a new stasis.
Economic theories often describe the behavior of entire markets or economies fairly accurately, but success in business is almost entirely dependent on details that are hidden when economists average across all sellers (competitors) and buyers (customers). Business thrive by exploiting imperfections in the market (see Arbitrage) and exploiting behaviors of customers and competitors that economists consider irrational (That is, not conforming to their theory. See Biases, Story and Homo Economicus.)
See also Value.