I've read about them both and looks like each has its advantages and disadvantages. I'm interested in your personal experience. Thanks.
I think a degu is easier to take care of. My aunt has had both of them and she says degus are less demanding. They shed less and are very clean. You won't have to clip their nails nor bathe them. All they need is a bowl with clean water and they'll bathe themselves.
Chinchillas can bite when they are not fully used to you. Although degus can bite too.
Chinchillas are less "fragile" in case you want to pet them.
Do you have kids? In this case I'd choose a chinchilla.
Neither should be getting wet. Both need dust baths which can get messy. In my experience degu's can be voracious chewers and love to burrow (kick shavings etc all around too!! ) Degu's are a little smaller so size wise are better although they still require a fair bit of space. The less cage space they have, the more time they require out of cage. Even with a sufficient sized cage they should still have at least an hour daily outside of the cage to bond with you and to run.

The biggest difference between a chinchilla and a degu boils down to two things for me. Chinchillas are more fragile and you have to worry about their fur slipping (although degu's tails can break off too...never grab a goo by the tail) Then their dietary needs...chinchilla food is much more readily available than degu food and it is inadvisable to give degu's anything but specifically formulated SUGAR FREE degu formula pellets as they are highly susceptible to diabetes. I personally have a lot of luck with degu's and when properly socialized they're really sweet and cuddly. Chinchillas are really interesting too but since they require more space than I currently have I haven't lived with one.