The essential difference is that in Buddhist meditation, at some point one uses one's mind to actively examine and investigate one's direct experience and to realize it's nature that way. This is the seventh factor of the Noble Eightfold path, Right Mindfulness, Samma-Sati. It is described in the Sattipatthana Sutta like this:
"There is the case where a monk remains focused on the body in & of itself — ardent, >alert, & mindful — putting aside greed & distress with reference to the world. He remains >focused on feelings... mind... mental qualities in & of themselves — ardent, alert, & >mindful — putting aside greed & distress with reference to the world.
In this way he remains focused internally on the body etc... in & of itself, or >externally on the body in & of itself, or both internally & externally on the body in & >of itself. Or he remains focused on the phenomenon of origination with regard to the >body, on the phenomenon of passing away with regard to the body, or on the phenomenon of >origination & passing away with regard to the body. Or his mindfulness that 'There is a >body' is maintained to the extent of knowledge & remembrance. And he remains independent, >unsustained by (not clinging to) anything in the world. This is how a monk remains >focused on the body in & of itself."
(Source: http://www.accesstoinsight.org/tipitaka/mn/mn.010.than.html )
The result of which is that the mediator begins to discern that reality is impermanent, unsatisfying, and non-self. As one realizes this more deeply, one is lead to liberation. This process is described in many places in the Suttas, most famously in the Anatta-Lakkhana Sutta like this:
"Seeing thus, the well-instructed disciple of the noble ones grows disenchanted with >form, disenchanted with feeling, disenchanted with perception, disenchanted with >fabrications, disenchanted with consciousness. Disenchanted, he becomes dispassionate. >Through dispassion, he is fully released. With full release, there is the knowledge, >'Fully released.' He discerns that 'Birth is ended, the holy life fulfilled, the task >done. There is nothing further for this world.'"
(Source http://www.accesstoinsight.org/tipitaka/sn/sn22/sn22.059.than.html )
Non-Buddhist systems of meditation often focus on just developing concentration without this process of analyzing direct experience, and so end up attaining other things, such as mastery of the Jhanas, rebirth into the Brahma worlds etc... There is nothing wrong with these other things, and they can also be a part of Buddhist practice, but they aren't the goal.