The answer is "Yes". You need both to really capture the moment. :{)#
As you're seeing, getting a good exposure isn't enough; you also need to express your view through the image's composition. But just being able to compose a shot won't always be enough; often you need to understand the impact that aperture and shutter speed (even though the camera will try to select them for you) make on the end result.
One thing that can help is studying the work of a number of photographers you admire. I found reading everything I could from the late Galen Rowell really helped me see more clearly, even when I wasn't shooting his typical subjects. Likewise, studying Elliot Erwitt's photos had a real impact. For the technical side, there are a number of authors; John Hedgecoe is good and has been around for quite a while; Bryan Peterson's Understanding Exposure is popular here.