First, intelligence (as a notion in psychology) is a theoretical construct... with numerous definitions. Wikipedia's article on intelligence lists a lot of them. Just going with the fist one there (which comes from an op-ed signed by many intelligence researchers), intelligence is defined as
A very general mental capability that, among other things, involves the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly and learn from experience. It is not merely book learning, a narrow academic skill, or test-taking smarts. Rather, it reflects a broader and deeper capability for comprehending our surroundings—"catching on," "making sense" of things, or "figuring out" what to do.
So, at a theoretical level, problem solving is seen a specific (but pretty important) aspect of intelligence. In practice, the way intelligence is usually tested in IQ tests, involves solving some types of problems (assuming a generous defintion of "problem"). Which essentially entails operationalizing intelligence. It's fair to say IQ tests test some specific types of problem solving. Of course the hope is that these correlate well enough with "problem solving anything".
There have also been some definitions proposed in psychology literature for what "problem solving" should mean. I'm not sure any of them has gained enough acceptance to be worth mentioning here.
IQ tests are however not the only investigated paradigm covering problem solving. There's a lesser known one that talks about CPS ("complex problem solving"), proposed in one definition to be:
“(…) the successful interaction with task environments that are dynamic
(i.e., change as a function of the user's interventions and/or as
a function of time) and in which some, if not all, of the environment's
regularities can only be revealed by successful exploration and integration
of the information gained in that process.”
However it's not terribly clear how that is different from the theoretical notion of intelligence. What is different is how [C]PS is operationalized, i.e. the tests used to measure it... and that actually affects the level of correlation:
This theoretical ambiguity is reflected in empirical findings on the
relation between CPS and intelligence. Multiple early studies indicated
that, while performance in CPS tasks varied tremendously among individuals,
psychological assessments of general intelligence were unable
to explain this variability (Brehmer, 1992; Rigas & Brehmer, 1999).
Kluwe,Misiak, and Haider (1991) summarized 11 of these early studies
on the relation between CPS and intelligence and concluded that most
of them failed to show a close relation between intelligence scores
and CPS performance measures. This led several researchers to suggest
CPS to be a cognitive construct mostly independent from intelligence
(Putz-Osterloh, 1985). Rigas and Brehmer (1999) summarized this
view in the different-demands hypothesis. To explain the weak correlations
that researchers observed between measures of general intelligence
and CPS performance, this hypothesis suggests that CPS tasks
demand the performance of more complex mental processes than intelligence
tests [and they mean IQ tests] do, such as the active interaction with the problem to acquire knowledge on the problem environment, which, in turn, results
in low empirical correlations between the constructs.
In reaction to these problems, Funke (2001) introduced Linear Structural
Equation systems (LSE) and Finite State Automata (FSA) as formal
frameworks that allowfor the description of underlying task structures.
Both of these frameworks enabled the creation of single complex systems [SCS],
which are independent of any semantic embedment (Greiff,
Fischer, Stadler, &Wüstenberg, 2014) as they only specify an underlying
system that can be clad in multiple semantic contexts.
In particular, the LSE formalism has been widely adopted by CPS research
and has led to the development of a considerable number of single
complex systems (e.g., Multiflux, Kröner, 2001; FSYS,Wagener, 2001).
In a further advancement, after Leutner, Klieme, Meyer, and Wirth
(2004) had used a combination of two single complex systems for measuring
CPS, Greiff, Wüstenberg, and Funke (2012) used the LSE framework
for the development of the multiple complex systems (MCS;
Greiff et al., 2014) approach, which was featured in the Program for International
Student Assessment (PISA) 2012, the arguably most important
large-scale assessment worldwide. [...]
The comprehensive answer to the question on the relation between
CPS and intelligence however, appears to depend on the
operationalization of CPS.Whereas themoderator analyses did not indicate
significant differences between measures of general intelligence
and measures of reasoning in respect to their relation to measures of
CPS, there are substantial differences inmean effect sizes found for studies
using different operationalizations of CPS. The smallest average effect
size for the relation of CPS and intelligence was found for classical
measures of CPS, M(g) = .339, followed by single systems based on
LSE, M(g) = .471. CPS scores gained from MCS tests are related most
strongly to intelligence, M(g) = .585 [...]
However, unlike classical measures of CPS or SCS, current MCS tests
do not feature some highly complex elements of problem solving such
as the recognition and handling of time-delayed effects. Thus, the cognitive
demands posed by MCS tests are likely to be relatively closer to
those posed by intelligence measures. Following the different-demands
hypothesis (Rigas & Brehmer, 1999), this might be causing the high correlations
of intelligence and CPS scores obtained from MCS tests. In
order to test this hypothesis, it would be necessary to develop MCS
tests that feature highly complex elements while simultaneously maintaining
high levels of reliability.
The bottom line is that there are problem solving abilities not entirely captured in basic IQ tests. Whether that tells us anything about "intelligence" vs "problem solving" is a matter of how one construes these notions theoretically.
Also asking if "Is [it just that] intelligence [is] learnt?" is a pretty different question, and I will not devote too much space to that here to that.
Of course intelligence (as usually measured, IQ) develops over time in adolescence, and according to some empirical stuedies the more schooling a child has, the better the outcome for IQ tests as well.
But this raises the question: what allows one to learn (better)? The answer, according to some is that there's no distinguishable factor between intelligence and learning ability. So basically, the question is circular in this view: you learn (better) because you are (more) intelligent, which make you (even more) intelligent. And that seems a pretty widely held view:
However, if we think of IQ as prediction devices, there is no better predictor of future learning than past learning. Furthermore, past learning does not just predict future learning—it often enables it.
But that hinges on the definition of intelligence as being this whole encompassing thing, which includes learning ability. Just going back to the opening definition of intelligence, but emphasizing a different portion of it should be enough to illustrate this:
A very general mental capability that, among other things, involves the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly and learn from experience. It is not merely book learning, a narrow academic skill, or test-taking smarts. Rather, it reflects a broader and deeper capability for comprehending our surroundings—"catching on," "making sense" of things, or "figuring out" what to do
It would make a good separate question whether learning ability and problem solving (rather than intelligence) can be substantially differentiated in cognitive tests... and whether assuming such a testing separation is possible, are the results highly correlated anyway?