Contemporary and classical ballet have often been compared to one another as being mostly similar, but one is ‘freer’ than the other. I think there are a lot more differences than that.
Although in both styles of dance, you need to pull up and ground yourself at the same time, contemporary dance is more ‘grounded’ whereas classical ballet is more ‘uplifted’. I would say for contemporary, you have to feel that 70% is grounded, and 30% is uplifted, but for ballet, 30% is grounded and 70% is uplifted. Stereotypical contemporary is mostly associated with a lot of rolling around on the floor, the music being slow, the atmosphere heavy, but really contemporary could be light too, just like classical ballet can be dark and heavy and not-always-fairylike. That’s why its hard to categorise which one is more grounded and which one is more uplifted – it really depends on the choreography.
For video references, this is a contemporary that focuses on being grounded and connecting with the floor even though there are a few jumps: https://www.youtube.com/watch?v=Mz07DQg3ajk
This is a classical dance that focuses on having the ‘energy’ go up rather than trying to connect with the floor: https://www.youtube.com/watch?v=zV1qLYukTH8
But another difference may be that for classical ballets there is generally a ‘role’ you are playing and a storyline to go along with it, whereas in contemporary it’s more expressing a feeling and a state of mind rather than trying to convey a story. For example, the famous dance of the Black Swan – the black swan is trying to tell the audience that she is an intruder in the prince and white swan’s relationship, and she is about to ruin the white swan life. However, in most stereotypical contemporary variations, there may be a person conveying the feeling of sadness and loss, or joy and hope.
As I have said before, these differences really depend on the choreographer, but I hope that this has cleared the differences up a bit for you!