Seeing happiness as feelings of pleasure does have distinct advantages. A feeling of joy is easily recognizable, oftentimes even to an outside observer, certainly to the person feeling it. It is the immediate, enjoyable feeling that is a main motivation for all (more highly developed) living beings. It comes straight out of the biological necessity for doing what is necessary to survive – eat, reproduce – and avoiding what is dangerous, causing pain.
For science, it is a good thing because there are observable, measurable components to it. One can even observe how the body and brain act when someone feels this kind of happiness; and few would doubt that someone who says that he or she is now feeling happy, joyous – enjoying a treat, being in a lover’s embrace, having won a contest – would be wrong about their feelings. Experiments can easily be constructed, too: Let somebody find a few bills of money, and their whole outlook on life will (for a little while) turn more positive. It is a good scientific approach to something so subjective, with the added advantage of avoiding judgment on moral grounds, having less interference from cognitive processes than many other aspects of life (at least at that particular moment in time, while it is being experienced).
Simple pleasures should not be underrated. The whole enterprise of a hunt for happiness is something of a luxury, after all. It only becomes possible to engage in it so strongly when the everyday joy of having shelter, food and social contacts is appropriately fulfilled, and conditions therefore offer the leisure time to wonder if that is really all. Even then, though, the pleasure of a good meal continues to exist. It only becomes less special when food is not a noteworthy issue, but it may also become something more, as learning and social status add new levels of symbolic meaning to what we choose to eat. Then again, when you are in a bad situation, you would probably look for the ways to gain a little happiness, and to get out of that situation into a better one.
The trend/wish in modern society is also to follow happiness as pleasure, being most concerned about feeling good, experiencing joy – and right now, at that. Unfortunately, this hunt after a quick fix is a rather infantile approach to life. If you don’t think much about the consequences, random sex or the hit of a drug will work well in giving quick joy, unfortunately. The sum of those moments of happiness is unlikely to provide a lasting happiness, however. What we tend to want, in the long term, is something more. We are not just shaped by our immediately experienced emotions, after all, but also by our thoughts and memories. And happiness, over the longer term, is not the same as the experience. Memories and interpretations – the stories we tell (at the very least, ourselves) about the course of our lives – play a major part. It may not have to go so far as the ancient Greeks suggested, for whom you could only judge the happiness of a life after said life had ended (after all, otherwise, there could still be a terrible turn of events). In that, things get more complicated than how we feel right now, but also much more into what makes us human.