Why do animals only eat some parts of their food?
It all depends on hunger. As in comment (@Remi.b), it has been mentioned, an animal won't throw if they are starving.


  For example monkeys/apes only eat part of a fruit and then throw the rest.


Monkeys and apes can differentiate between good and bad part of the fruits. So, they eat what they can and if the color looks different (or however they differentiate them) they will throw the rest of them. 


  Cats (big and small species) only eat some parts of their prey and then they abandon it.


Cats in general are very picky eaters. They actually have a very short digestive tract, so, they need something easy to breakdown into energy rather than complex processes of digestion. Their main food consist of meat, they leave the inner organs as such. 

Source:


http://en.wikibooks.org/wiki/Wikijunior:Big_Cats/Complete_Edition
http://bigcatrescue.org/feed-cats/ 






Image Source:


http://wordpress.as.edu.au/rspiers/2013/05/22/comparing-animals-digestive-systems/
http://cats56971.wordpress.com/2012/10/30/eating-habits-of-a-african-lion/