The enemy-centric approach basically understands counter-insurgency as a variant of conventional warfare. It sees counterinsurgency as a contest with an organized enemy, and believes that we must defeat that enemy as our primary task. There are many variants within this approach, including "soft line" and "hard line" approaches, kinetic and non-kinetic methods of defeating the enemy, decapitation versus marginalization strategies, and so on. Many of these strategic concepts are shared with the population-centric school of counterinsurgency, but the philosophy differs. In a nut-shell, it could be summarized as "first defeat the enemy, and all else will follow".
The population-centric approach understands counter-insurgency as fundamentally a control problem, or even an armed variant of government administration. It believes that establishing control over the population, and the environment (physical, human and informational) in which that population lives, is the essential task. Again, there are many variants within this approach, including some very hard-line methods and some softer approaches, but the underlying philosophy is "first control the population, and all else will follow".
(Note that we're talking about classical counter-insurgency theory here, not modern counter-insurgency practice, so much. Also, I'm not suggesting one school is always right and the other always wrong -- both can be well-done, and both can be hopelessly counterproductive if done badly. The key to "good counterinsurgency practice" is the agile integration of civil and military measures across security, economic, political and information tracks -- and this is something that has to be done regardless of which approach you adopt, and is just as necessary in both).
Now, some people are quite committed to one or the other school of thought (Galula, for example, flatly states that the population-centric approach is always correct, and the new FM 3-24 takes a similar but less absolute stance). But my experience has been that both are applicable in varying degrees in most insurgencies, and at different times in the life of any one insurgency - since, over time, the nature of insurgencies shifts.
The real art is to "read the battle" and understand how it is developing, fast enough to adapt. Neither the enemy-centric nor the population-centric approaches are always or universally appropriate -- there is no cookie-cutter, and no substitute for situation-specific analysis informed by extremely deep local area and cultural knowledge.
As an example of the need to read the battle and adapt, I hope you will forgive a brief personal anecdote. In Timor in 1999 I worked closely with village elders in the border districts. I sat down with several of them one afternoon to discuss their perception of how the campaign was progressing, and they complained that the Australians weren't securing them in the fields and villages, that they felt unsafe because of the militia (the local term for cross-border guerrillas) and that we needed to do more to protect them. In actual fact, we were out in large numbers, securing the border against infiltration, patrolling by night, conducting 14 to 21-day patrols in the jungle to deny the militias a chance to build sanctuaries, and working in close in the villages to maintain popular support. There had not been a single successful attack by the insurgents on the population for more than two months. So, "objectively", they were secure. But -- and this is the critical point -- because our troops were sneaking around in the jungle and at night, staying out of the villagers' way and focusing on defeating enemy attempts to target the population, they did not see us about, and hence did not feel "subjectively" secure. This was exacerbated by the fact that they had just experienced a major psychological trauma (occupation, insurgency, mass destruction and international intervention) and as a society they needed time and support for a degree of "mental reconstruction". Based on their feedback (and that of lots of other meetings and observations) we changed our operational approach, became a bit more visible to the population and focused on giving them the feeling, as well as the reality, of safety. Once we did that, it was fine.
In other words, we had to shift from a more enemy-centric approach to a more population-centric approach to adjust to the developing situation. My personal lesson from this experience was that the correct approach is situation-dependent, and the situation changes over time. Therefore the key is to develop mechanisms that allow you to read the environment, to be agile and to adapt, as John Nagl showed so brilliantly in Learning to Eat Soup with a Knife.
So, in summary, two broad philosophical approaches in classical counterinsurgency (and remember it's classical 20th century counterinsurgency we're discussing here) -- population-centric, and enemy-centric. Both have merit, but the key is to first diagnose the environment, then design a tailor-made approach to counter the insurgency, and - most critically - have a system for generating continuous, real-time feedback from the environment that allows you to know what effect you are having, and adapt as needed.