I'd say that the cheerful price is a primarily psychological concept, while shadow price is a more analytical one, and that is the whole point - when what you feel and what you think you ought to feel disagrees, the concept of cheerful price is explicitly telling you to not worry about the mismatch, and go with the former.

The Wikipedia page is not exactly easy to read for someone not already familiar with the term. My impression (possibly wrong) is that "shadow price" = estimate of a hypothetical market price, if the thing is actually not on the market (therefore it is impossible to look at the actual price).

So, first, there is a difference in usage. "Shadow price" is used in context of buying something for money, or producing it (which would cost you money and other resources). "Cheerful price" is used in context of selling, typically your work and time. In other words, you estimate "shadow price" of things you want, and "cheerful price" of things someone else wants you to do.

Second, there is a difference in precision. "Shadow price" tries to be as accurate as possible (in a situation where it is difficult to be accurate). "Cheerful price" tries to be safe. That is, if your probability distribution is like: most likely $200, but there is a small chance it is $100 or $300, you would choose $200 for the "shadow price", but $300 for the "cheerful price".

Suppose I have a state consisting of $(a,b)$: number of apples, number of bananas. My utility is $U_a\left(a\right)+U_b\left(b\right)$ - i.e. I have no terminal desire for money, and my utility for apples and bananas is separable (aka a sum, with each term dependent on only one of the two). I also have a budget constraint: $P_{a}a+P_{b}b\le m$, i.e. price of apples times number of apples plus price of bananas times number of bananas is at most $m$, the amount of money I start with.

A useful technique for this sort of problem is to separate it into two optimization problems (thus the term "separable" for the utility). Here's how that works:

• First, pick some amount of money $m_a$ to budget to apples, and budget the rest $m_b=m-m_a$ to bananas.
• Then, maximize $U_a\left(a\right)$ subject to $P_{a}a\le m_a$, and separately maximize $U_b\left(b\right)$ subject to $P_{b}b\le m_b$.

(Note: if $a$ and $b$ were higher-dimensional, i.e. each involved a whole bunch of variables, then this would still work, and indeed that's the case in which it's typically interesting/useful to do this.)

Assuming that each sub-problem can be solved efficiently, all we have to do is adjust the budget-split $(m_a,m_b)$ to be optimal. That's where the shadow price comes in: the shadow prices of each sub-optimization problem tells us how much utility we would have to trade off in the $a$-problem to get some amount of utility in the $b$-problem, or vice-versa. To put it differently: the shadow price summarizes all the information about the $a$-decisions which is relevant to the $b$-decisions. To put it yet another way: the shadow price is the opportunity cost of spending a marginal dollar on $a$ rather than $b$.

And this exactly the way people use shadow prices in practice, all the time. When I'm trying to avoid spending a lot of money on a new gadget, it's not because I value money as an end in itself, it's because I can use money saved to gain utility in other problems in my life. When trying to decide how much to spend on the gadget, I don't need to directly think about all the other problems in my life - I don't have to think about it all as one giant optimization problem. Instead, I just have a rough intuition for how much utility I can get in the other problems of my life from $X extra, and then I decide whether I can get more utility than that by spending $X extra on the new gadget. That rough intuition for how-much-utility-$X-is-can-buy is the shadow price.

Now we're ready to answer the question. The shadow price quantifies the opportunity cost, so if I'm paid my shadow price, then that's just barely enough to cover my opportunity cost. It's reimbursing me for the opportunity cost of doing-the-thing rather than spending my time and resources on other problems in my life, but it's just reimbursing me, nothing extra. I'm not actually eager to take the deal, I'm indifferent. 

The idea of a cheerful price is that it does give me something extra. It's not just paying my opportunity cost, I'm not just indifferent between taking the deal or not; it's giving me more utility than I'd get from spending my time and resources on other problems in my life.

The other answers are pretty great but I just want to point out that it very well could be the case that 'cheerful prices' are just a specific subset of 'shadow prices'.

The concept of cheerful price also seems aimed at a very different audience!

But translation is hard – I'm not sure the two prices are different 'fundamentally'.