When the term billet is used it generally refers to a cast piece machined out of a larger cast piece... not a sandcasted or form casted piece.
When you cast any metal there are alot of things that go on. When you have changes in geometry of your cast piece there will be weird temperature gradients and you'll end up with inclusions etc...
Long story short, you will have a product with superior properties (strength, ductility etc.) if you make one large cast piece where the bottom or center of it is guaranteed to be very homogenous metallurgical structure and then machine your part out of that area of the casting.
You don't really refer to steel parts like this as billet steel. Its more a term for materials like aluminum, stainless and other alloys. The advantages of this process aren't as noticed with plain low carbon steel. Like billet levers for motorbikes, billet clutch covers.. etc. A lot of aluminum is done this way.
Forged steel is a casting of steel that is heated up to high temperature and hot formed to whatever you want by stamping, rolling or other methods, no machining. You will get very good directional properties with a product like this, but there is very high residual stresses. So often, products will be forged and then heat treated to relieve stresses.
There is no such thing as a forged billet. You either make a billet part out of a casting by machining it. Or you forge the part out of the casting through hot working the casting.
Do to directly answer your question, a billet part is often of better quality and higher strength than an identical cast part. And a forged piece may be substantially better suited to an application, but that depends on the application.
In metallurgical terms, there often isn't one forming or finishing process thats the best. Each have advantages and disadvantages, and based on what the application is you'll know what properties you want the most and create the piece accordingly.