Are they one and the same, or do they have different meanings?
The two terms appear to be used without particular care on the show. Ultimately, they refer to the same thing--the hardware architecture of Data's brain, but they conceptualize it in different ways.
Neural nets and positrons (anti-electrons) are real things. A neural net is a kind of computing architecture that is, roughly speaking, modeled after the functioning of a living brain. Per Wikipedia:
Neural networks (also referred to as connectionist systems) are a computational approach which is based on a large collection of neural units loosely modeling the way a biological brain solves problems with large clusters of biological neurons connected by axons. Each neural unit is connected with many others, and links can be enforcing or inhibitory in their effect on the activation state of connected neural units.
So the "neural net" formuation is characterizing Data's brain in terms of its connection, function, and development structure.
In contrast, calling it a "positronic brain" is talking about it at a deeper hardware level. Positrons have no internal structure, so they must be acting as current/signal carriers inside the computing apparatus. Presumably, the nodes of the artificial neural net are each positronic components.