“We must distinguish between being non-scientific and being unscientific. [] philosophy [perhaps panpsychism and/or string theory] is a subject which is almost certainly of its very nature non-scientific. We must not jump from this purely negative statement to the conclusion that it has the positive defect of being unscientific.” — C.D. Broad

“‘Unscientific’ sounds more negative. It implies that something is pseudoscience. (A scam that is pretending to be science but not using the scientific method properly.) ‘Non-scientific’ is more neutral and might be used to describe parascience. (Fields of study that are not science or math but still valuable in their own way, such as law and history.).”
— See source here.

Let’s start off with a passage from the English philosopher C.D. Broad (1887–1971):
[As found in Broad’s paper, ‘Philosophy’, in Inquiry I.]
Interestingly, C.D. Broad had telekinesis, mind-reading and backwards causation in mind when he wrote the words above. [See this article on Broad in Psi Encyclopedia.] That alone shows us how problematic the distinction between x being unscientific and x being non-scientific is. What’s more, this distinction may allow too much leeway for the subjects under discussion (i.e., even when it comes to what’s non-scientific, never mind what’s unscientific).
In any case, if we forget telekinesis, mind-reading, etc., it’s feasible that both string theory and panpsychism can be non-scientific without thereby also being unscientific. However, there’s an obvious difference here. As stated, string theory is a science and panpsychism isn’t. Thus, if string theory is either non-scientific or unscientific, then that’s a problem. Perhaps panpsychism, on the other hand, is only problematic if it’s unscientific.
[I believe that it’s fair to say that the strong term “pseudoscientific” doesn’t apply to either panpsychism or string theory… Of course, some readers may disagree with that!]
So let’s speak in broader terms here.
Is Panpsychism Unscientific or Non-Scientific?

Panpsychism is a philosophical theory, whereas string theory is part of science. (Some — perhaps even many — physicists question the scientific status of string theory. See here.) Nonetheless, some of the philosophical (i.e., not scientific) problems which both face are similar.
Of course, panpsychist philosophers don’t claim that panpsychism is an actual science — or even that it’s scientific.
Philosophy itself isn’t a science. Consequently, that must mean that panpsychism isn’t a science.
(Some kinds of philosophy are classed as being “scientific”, without thereby also being actual examples of science. See here.)
Despite the words above, many panpsychists (as with various other metaphysicians) claim that panpsychism must still be respectful of physics. [See here and here.] Alternatively, they say that it shouldn’t (directly) contradict anything in science. Indeed, because panpsychism deals with “intrinsic nature”, then — almost by definition! — it can be said that it can’t contradict anything in physics.
To return to the difference between non-scientific and unscientific.
If we say that panpsychism is non-scientific (to quote Broad again), then
“[w]e must not jump from this purely negative statement to the conclusion that it has the positive defect of being unscientific”.
The last statement from Broad is very helpful in this context. Let me re-quote it:
“The latter term [‘unscientific’] can be properly used only when a subject, which is capable of scientific treatment, is treated in a way which ignores or conflicts with the principles of scientific method.”
This means that if panpsychism deals with a subject “which is [in]capable of scientific treatment” (and which “is [also] treated in a way which ignores or conflicts with the principles of scientific method”), then we may have a philosophical problem. The point here is that panpsychists will simply claim that panpsychism has nothing whatsoever to do “with the principles of scientific method”.
Some philosophers may also claim that panpsychism and science are two non-overlapping worlds or language games.
In detail. This parallels — at least to some extent — the Wittgensteinian claim that science and religion don’t contradict each other because they’re dealing with different phenomena.
This means that science and religion are — in the words of Steven Jay Gould — “non-overlapping magisteria”.
Alternatively, some philosophers may claim that science and religion sometimes deal with the same phenomena, though in very different ways.
On this Wittgensteinian view, then, religion is non-scientific, not unscientific.
So does this also mean that panpsychism and physics are also non-overlapping magisteria? So, if that is the case, then how can they contradict one another?
That said, I would assume that panpsychism surely can’t be in an equivalent position as religion vis-à-vis its relation to science (or to physics).
So some of the claims above may be too convenient for panpsychism.
Take the following position:
i) If physics states that there are no “intrinsic properties” (certainly if it claims that micro-entities don’t have “phenomenal properties”),
ii) then panpsychism does indeed “conflict with the principles” of physics.
The problem is, however, that panpsychists (as well as some physicists) claim that physics simply “ignores” intrinsic properties. In other words, physics has no position at all on such things. As the philosopher Philip Goff once put it:
“This [panpsychist] argument presses us to the conclusion that there must be more to physical entities than what they do: physical things must also have an ‘intrinsic nature’…”
Elsewhere, Goff wrote:
“[G]iven that physics is restricted to telling us only about the behaviour of physical entities — electrons, quarks and indeed spacetime itself — it leaves us completely in the dark about their intrinsic nature. Physics tells us what matter does, but not what it is.”
Some (perhaps many) physicists, on the other hand, will simply claim that intrinsic properties don’t exist. Thus, we can conclude by saying that physics as a whole — as stated — doesn’t have a universal or systematic position on intrinsic properties.
The other much broader point which has to be made is that the scientific criticisms of panpsychism can also be applied to many (actually, most) other philosophical theories. Indeed, the entirety of metaphysics has been said to be suspect from a scientific point of view — at least according to some scientists and philosophers at certain points in history!
All this means that panpsychism’s relationship with science is far from being unique.
Yet philosophy can indeed be unscientific, rather than simply non-scientific.
What about string theory?
Is String Theory Unscientific or Non-Scientific?

A simple — and perhaps naïve — example of the problem of string theory’s scientific credentials relates to the question as to what is and what isn’t observable and/or testable.
If what can be observed and/or tested — or can be in observed and/or tested (as it’s often put) “in principle” — is ignored or rejected, then surely that would be an unscientific position to take. However, even within science, many things are unobservable (e.g., quarks, protons, the iron core at the centre of the earth, distant galaxies, fields and forces, etc.). Some things are even unobservable in principle (e.g., the past, numbers, laws, universals, perhaps other minds, etc.).
All that clearly has relevance to both string theory and panpsychism.
Indeed, most (or all) of the claims of metaphysics (even those claims about things which are observable and/or testable) are still primarily about things which aren’t observable and/or testable.
This also means that if science has no problem with panpsychism (or even accepts the fruitfulness of research into it), then that would also fit in well with the “theoretical pluralism” referred to earlier.
Now what about string theory and the non-scientific/unscientific bifurcation?
We can say that because of the current state of play, many of the claims and theories of string theory aren’t (to use C.D. Broad’s words again) “capable of scientific treatment”. (Not only because of the experimental limitations brought about by the deficiencies and limitations of contemporary technology.) Nonetheless, that doesn’t necessarily also mean that string theory “conflicts with the principles of science”.
Of course, it can now be said that if string theory’s claims aren’t capable of scientific treatment, then how can they still be scientific? However, couldn’t this question also be asked of many 19th century and 20th century theories (in physics)? That is, many theories were created before the evidence, experiments or tests were in. So does that also mean that such claims and theories were unscientific?
What’s more, if the experiments and evidence did come later, then surely we can’t still say that such formerly untested theories were unscientific.