Is there any difference between 'all real numbers' and '$(-\infty, \infty)$' I've just thought about this.
All the textbooks I've been looking at for pre-calc, 
the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as '$(-\infty, \infty)$.
Is there any difference using the terms vice versa (implicitly)?

There is no difference: $\Bbb R = \left]-\infty,+\infty\right[$. Writing it like this serves to get you used with the symbol $\infty$, I guess (mostly psychological reasons?). Also, there will be a time when you'll need to use concepts dealing with the extended real line, so it will be natural to talk about: $$\left]-\infty,+\infty\right], \quad \left[-\infty,+\infty\right[, \text{ and } \left[-\infty,+\infty\right].$$
