The basic difference between Buddhist meditation (BM) and secular mindfulness practices, like Mindfulness-based Stress Reduction (MBSR) is the objective of practice.
In both practices, the foundation is present moment awareness; however, in most secular mindfulness practices, awareness in and of itself is the goal—a sort of mental stability and groundedness. When we are aware, we are not lost in mental-emotional chatter (aka the proximate cause of most stress). In turn, we are able to stay more focused and in alignment with our deeper values.
Inversely, BM uses goes beyond mental stability and uses that awareness as a foundation to learn about the nature of body and mind—for example, when anger arises, what does that feel like? What sorts of thoughts and sensations are connected? How long does it last? What precedes it? What follows it? Is the experience of it pleasant or unpleasant? Most importantly, what are its roots? These are not questions to analyze with thinking, they are questions to explore with awareness as they are happening.
The effect of all this learning is the development of understanding or wisdom. When, on a very deep level of mind, we know that McDonalds isn’t good for us, we simply stop eating McDonalds; likewise, when we know to a similar depth that anger is not good for us, or that indulging in mental-emotional chatter is not good for us, we simply stop doing it—stress reduces and positive mental qualities increase.
The base is the same, but the difference is that the aim of Buddhist meditation is a bit grander; when the weeds of mind arise, MBSR cuts them off above the soil, whereas BM yanks out the entire root.
If the Buddha spoke in the contemporary lingo, he may very well have named his system mindfulness-based wisdom cultivation.