What is the difference between solving a problem and acquiring a skill?

To take your concrete examples, there are several broad distinctions relevant to comparing a task like chess and a task like playing Pac Man.

Cognitive versus psychomotor skill

• Research on learning and task performance is divided into various domains of tasks. Two such domains are cognitive, of which chess would be an example, and psychomotor, of which Pac Man would be an example. Video games vary in the relative importance of cognitive versus psychomotor skill.

• You might want to read the Annual Review of Psychology article by Rosenbaum, and Carlson, and Gilmore, 2001, PDF "Acquisition of intellectual and perceptual-motor skills". The article provides an overview of the similarities and differences between cognitive and psychomotor skill acquisition.

• Traditionally, the word "skill" has had psychomotor connotations. But ultimately the defining feature of a skill is that it is acquired through practice, and this applies well to cognitive as it does to psychomotor tasks.

Learning versus performance

• Some research focuses on how learning occurs, whether it be studying the relationship between practice and performance, examining changes in the errors made over time, and so forth (e.g., see this review by Heathcote et al PDF, called "The power law repealed: The case for an exponential law of practice" which is on the mathematical functional form of the learning curve).
• Other research focuses on performance itself. What are the cognitive processes that underlie performance? In such studies, learning is often a nuisance factor.
• Thus, in the case of both Chess or Pac Man you could focus on either the process of learning or on performance itself (whether it be of a novice or expert).

Task complexity and time frames of learning

• Research in skill acquisition can broadly be distinguished in terms of the typical time frame it takes to acquire the skill of interest.
• Typical skill acquisition research that is done in the lab involves tasks where meaningful learning can be observed in hours or perhaps days.
• Some other skills can take years to acquire, and this is broadly studied in what is sometimes referred to as the "expertise literature". There have been many studies of chess within the expertise literature. See for example this paper on "the role of deliberate practice in chess expertise" PDF by Charness, Tuffiash, Krampe, Reingold, & Vasyukova, (2005).
• With regards to this distinction, Pac Man presents some difficulty. No doubt one could study meaningful learning in the short term, but also that it might be possible to focus on the learning that might occur for a true Pac Man master who had devoted years to perfecting the art of escaping ghosts and eating little pellets.