The definition of a packing measure in Falconer's Fractal geometry is given by
I am assuming that $\mathcal{P}^s(F)$ as defined in 3.24 is an outer measure (this is not stated in the book).
Now in Mattila's Geometry of Sets... they use $\textbf{closed}$ disjoint balls in 3.22 in Falconer. I dont think this makes a difference but in 3.24 they have defined the measure (which in Mattilas book means outer measure) with $ F=\bigcup_i F_i$. Does this make a difference? Does any theorem on packing (outer) measures work either definition?.
Yes - you are correct on both counts. The function ${\cal P}^s$ is an outer measure (in fact, a metric outer measure) and its value is unchanged if we used closed balls or open balls in it's definition (at least, not in Euclidean space).
If you are interested in these sorts of issues, you might consider looking at the book Measure, Topology, and Fractal Geometry by Gerald Edgar. Falconer's book is very good but, also, very broad. As such, it deals with measure theory on somewhat intuitive level, necessarily glossing over points such as those in your question. Edgar's book, by contrast, is much more focused on measure theoretic details. The fact that ${\cal P}^s$ is a metric outer measure, for example, is proved as theorem 6.2.7 of the second edition of the text.