All things being equal, a much larger EPS should tend to make many opportunities possible, including better financial aid, higher faculty salaries, and more ambitious building programs. The difficulty is, all things are almost never equal. No two universities are likely to be equally adept at managing their endowments, nor are they likely to face identical local costs or incur sudden major expenditures at the same time. Some endowments are so huge that they must be generating surplus revenue that is simply plowed back into the endowment, thereby resulting in little or no noticeable improvements to student life (compared to schools with endowments that are a little less huge.)
I think one would need to dig deep into comparative costs and expenditure data to even begin to get a handle on how endowments affect student opportunities. Nevertheless, in general, schools with the highest EPS numbers tend also to be among the highest-ranked, most selective in America. More money tends to buy more stuff, and better stuff. On the other hand, some colleges will be better than others at making a little go a long way.