When we compare the difficulty level of various problem types, we are comparing two things:
The easiest problem types for children to understand and solve are the Join and Separate Result Unknown. These are easiest because there's a time-delineated story that tells them what to do to act out the problems with manipulatives. When children try to solve these problems by counting up or down then the Join problems are significantly easier because counting forwards is easier than counting backwards, but the acting-out (direct modeling) methods of solving those problems are pretty much equally easy.
The next easiest problems that children learn to understand are the Part-Part-Whole, Whole Unknown problems. These are solved in the same way that Join Result Unknown problems are solved, so that makes them easy. The one difficulty that some children have with the Part-Whole problems is that some children don't understand the question--that you want to put two groups together and count them as one group. Once they understand the question, these problems are ones children understand and solve well.
These first three types of problems are often included in kindergarten math goals and standards (with restrictions to suitably small numbers).
The next set of problems children learn to understand and solve are typically the Join Change Unknown problems and the Compare Difference Unknown problems. These both have direct modeling strategies that make sense to most children and that many children will invent if they are given appropriate problems and opportunities. So far as I know, children don't necessarily understand and learn how to solve one of these earlier than the other. I know there are children where the compare question (how many more or how many fewer) is a tricky one that they don't completely understand, and it seems that showing a direct modeling/matching picture is a good way to explain what the question is asking.
Separate Change Unknown problems are significantly harder than Join Change Unknown problems because direct modeling involves either counting back (harder) instead of counting on (easier) or recounting the remaining set, possibly several times (guess and check). Some children will figure out this direct modeling strategy, and for them Separate Change Unknown problems will become ones they are confident with solving. Other children won't figure this out, and will learn how to solve these problems by relating them to Part-part-whole Part Unknown problems
Part Part Whole, Part Unknown problems are ones that children learn how to solve by relating them to easier problem types. In the problem:
Mike has 4 red marbles and some blue marbles. He has 6 marbles all together. How many blue marbles does he have?
One way a child might solve it is to put out 6 counters (for the total 6 marbles) and then make a subset of 4 (to show the red ones) and count the remaining ones (which represent the blue ones). Doing it this way shows the connection between part part whole-part unknown problems and separate result unknown problems.
Another way children might solve the problem is to put out 4 counters (for the red marbles) and count on to 6 (adding 2 more counters to the pile as the blue marbles). Doing it this way shows the connection between part part whole, part unknown and join change unknown.
Both of these ways of thinking about the problem work, make sense, and give the child a way of direct modeling to solve part-part-whole, part-unknown problems that connects to their prior knowledge.
Compare, Compared Quantity Unknown problems aren also ones that children learn how to solve by relating them to easier problem types. In the problem:
Mike has 4 marbles. Steve has 2 more marbles than Mike. How many marbles does Steve have?
We know that make has 4, so Steve has 2 more than 4. Figuring out 2 more than 4 is something that connects pretty easily to having 4 and getting 2 more: a Join Result Unknown structure.
Mike has 4 marbles. Dan has 1 fewer marbles than Mike. How many marbles does Dan have?
We know that make has 4, so Dan has 1 fewer than 4. Figuring out 1 less than 4 is something that connects pretty easily to having 4 and giving one awa: a Separate Result Unknown structure.
So these second two problem types: PPW-PU and CQU are not too hard because it's not too hard to connect them to earlier easier problem types.
Join and Separate Start Unknown problems are both best solved by working backwards from the result number to the start number. This means that the child needs to be able to translate a change-statement saying: "Mike got 2 more marbles" into "Mike had 2 fewer marbles before". Working backwards is a more complex version of making connections than we needed in the previous problem types, so these are harder.
Compare Referent Unknown problems are also working backwards kinds of problems. You are given information like: "6 is 2 more than what Steve had", and you have to be able to turn that around to think that means that Steve has 2 less than 6. Turning around a comparison statement makes the process of solving a CRU problem more abstract and difficult than solving a CQU problem.
Only the most basic problem types are included in the kindergarten curriculum. This means Join and Separate Result Unknown problems, Part-Part-Whole Whole-Unknown problems, and sometimes Part-Part-Whole Part-Unknown problems. If part unknown problems are taught in kindergarten, they are typically taught as a process of direct modeling by the Separate, Result Unknown strategy.
The Common Core Math Standards explicitly state that students in first grade should be solving problems of all of these problem types, and that standard is repeated again in second grade. If you look at current textbooks, you will see that most of them do not include the full range of story problems. It will be interesting to see how/if things change...