Why do full service airlines prefer to run a separate low-cost carrier?
They still want to get into the low cost segment but want to market themselves differently for better penetration. You can only present so many images of a brand to customers: You can't show both the super-cheap prices with happy youth on them and posh first class at the same time.
They do not want to tarnish a nice brand by mixing it with a low cost carrier and what that is perceived to entail. For instance, Germanwings got a somewhat infamous reputation recently, and Lufthansa is probably happy their name was not more frequently used in media. Low cost is also associated with less good customer service and hidden fees.
Crew are often less well paid, and they are sometimes put under a separate working agreement and company to try to prevent conflicts. Some legacy carriers have very good pay which is difficult to reduce. Management does not want to be bound by these when expanding or adjusting their business.


Each case is pretty specific depending on exactly they are trying to do and achieve. For many American Carriers and Lufthansa, the last one is probably the principle reason.