While many blogs have already discussed and dissected what both 'character skill' and 'player skill' are, lets look at why they're different and why they're also used differently.
As most that play just about any version, mutation or blatant rip of D&D know, in the beginning the game gave characters statistics of their raw potential, some perks and abilities that differentiate the classes, and very specific character skills for the thief. Beyond this more rules-light appearance, characters would go on adventures, relying heavily on the goodwill of the dice gods and the mental skills of the player, and hopefully wouldn't die as a result. This approach let the faculties and imaginations of the players steer their destiny, and often led to moments of "outsmarting" or "besting" the DM and his well-thought-out encounters and puzzles. It also lead to a large number of retarded things that players have become infamous for to those inside and out of the hobby.
On another end, we encounter the later editions that bring new light and focus on character skills. This focus became so acute that eventually the rules systems themselves became intricate and complicated, pushing out player skill, where suddenly the character needed to invest time and thought to allow themselves certain abilities not readily available at level one. Suddenly Tarzan needed to take the feat Brachiation just to get around, and even then he had to roll against a difficulty class, or hit the nearest tree trunk. Whether intentional or not, the realm of pen & paper seemed to be turning the way of the new-wave video game role-play.
So, do you find one to be better than the other? Regardless of what side of the fence you lovely readers may sit, I will take a sledgehammer to said fence. Both of these approaches have their uses and the most obvious use, to me, is the perceived value of education present in the game, and its setting. When playing D&D I enjoy knowing I won't be made to roll for every 5 feet of rope I climb, for fear of the eventual and impending critical failure of a '1' of the d20. And when playing Star Frontiers I'm without worry of the percentile roll for one of my character's skills; it's easy and straight forward. Both of these settings mentioned and the skills presented cover two different areas; climbing rope is not beyond anyone with bodily strength, but writing a program into a science fiction computer requires knowledge and training, where still yet such a task comes with the chance of human error. Even without rush it can be as simple as one too many letters in a string of code that crashes the system.
So what do I think? A proper balance of both player and character skills makes for a good system when it pays attention to the setting. Characters are only as perceptive as their players are because they rely so heavily upon direction, while the character may be more fluent in conducting major surgery on alien species that the player could not even begin to comprehend their anatomies. Dungeons & Dragons is (generally) placed in a realm where knowledge is as primitive as medieval (or renaissance) times, where much went to faith, and common sense was the measure of the day, and is a skill that only players bring to the table. Star Frontiers is a universe where technologies easily surpass our modern world (zeerust aside) and the characters themselves bring the skills to the table. This is why they play different, but still both utilize some of one and more of the other category of skills.