What Problems Are Applied Mathematicians Concerned With?
I am mostly going by your title question and your clarification from the comment. Because as it turns out, the only field most applied mathematicians add knowledge to is applied mathematics itself, only then someone else to apply this knowledge to something later, so there is a lot of internal stuff going on. But more about that later. First let me tackle the part of applied mathematics that is "exposed" to the sciences. (By which I choose to mean everything except maths for sake of simplicity)
What do applied mathematicians do with respect to sciences?
If you want to apply mathematics to something, there are essentially two (oversimplified) steps:
- Write down a model of your problem. (e.g. differential equations, graphs, minimization problems, etc.)
- Extract useful information from your model using maths. (e.g. find solutions, universal properties, minimizers, numerical approximations etc.)
People from the sciences¹ are certainly good at the first step, and they are taught a bit about the second step during their education or pick something up in practice. But that is generally restricted to some standard methods. In contrast think of applied mathematicians as kind of the opposite of this. The model a mathematician writes down might be a bit rudimentary and not be most realistic one, but he'll certainly know some more advanced mathematical trickeries to extract some additional predictions from it. And of course as lifetimes are limited, it is basically impossible for someone to get into the full depths of both sides.
Now one would think that this would lead to a clear division of labour. But what muddies the waters is that there are many models and not all of them are created equal. There are many models that may look reasonable but that are mathematically ill-posed, rendering their predictions useless.² There are also many models that look simple and are of interest to scientists, but where extracting mathematical information is impossibly difficult, while a less straightforward looking model might be easier to solve. (And for the converse, there are many mathematical toy-models with interesting theorems but no real applications).
People who focus on this would generally fall under the heading of mathematical modelling. But that is only a small part of applied mathematics.
What problems are applied mathematicians concerned with?
Contrary to your question, a lot people in applied mathematics don't deal with specific real world problems at all. Rather than that they work on improving the methods used for problems. These can be ways to prove certain classes of theorems and make predictions or they can be algorithms to numerically solve problems. Both are completely different but kind of have the same function for the sake of this discussion.
The important point is that the good methods are application agnostic. I don't want to prove a new theorem for each new application and I don't want my numerical algorithm to only work for a specific equation (there are some exceptions but those are by necessity, not by choice). So while many may look to some application as an inspiration or put it in the paper as an example, that is not their focus.
Notes:
¹As a sidenote, you called them "domain specialists", but that isn't a good distinction, as this is true for both sides. Applied mathematics splits in many domains and each mathematician will only be a specialist in one or two of them (e.g. statistics, or partial differential equations or, certain numerical methods etc.). They just might feel like "jacks of all trades" to you, because each of those domains will have some applications in almost all sciences, but they are still rather specific and fundamentally useless if the problem needs a different subfield of methods. Think of the two as orthogonal to each other.
²If a model M has no solutions, the sentence "Every solution to the model M fulfills property X" is true for any X.
Broadly speaking, a domain expert knows what questions to ask, and an applied mathematician knows how to get useful answers once the question is posed, because they are familiar with common patterns that repeatedly appear in the structures of questions about how the world works. Good progress can be made when they work together, leveraging their combined skills.
There is of course a continuum here with many intermediate stages, such as modeling experts, who specialize in abstracting domain problems into mathematical form, and so have some breadth in both the domain and applied mathematics, and so form a bridge between deep domain experts (who have specific goals they are trying to achieve) and the applied mathematicians (who have the tools to fully exploit the models, and are familiar with edge-cases and other problems that may arise during analysis).