Contemporary in some kind of way expresses the present, What happens right now in this specific timing. Thats exactly what contemporary dance does. The dancer is able to express many feelings with their body making the spectator feel exactly what they are trying to say.
When we hear the word Dance, many of us think of Ballerinas. Sophisticated and elegant but when it comes to contemporary dancers, the elegance is lost and the priority is the rawness of the movements like in the picture we see underneath. 
No shoes or tutus that is what is all about when it comes to contemporary dance. A mix of ballet, jazz and personal touch makes it all come together as a different dance. So it is definitely worth it a look, it gives you goosebumps and actually feel and understand the story the dancer is trying to tell.
Image: http://swilkinsdance4u.wordpress.com/2012/10/12/contemporary-dance/
I know many dancers that have said similar things. Whereas classical ballroom dance such as ballet trains its dancers to be the same and have perfected existing movements modern dance likes to take a turn and go for originality over being the same.
I really liked how you differentiated between contemporary and ballet. I think the raw and natural feel of contemporary dance as you described it is a sound departure from more “traditional” dancing.