What are some of the privileges that the Inner Party members possess that the Outer Party members don't have?

The Inner Party elite live lives of luxury by comparison to everyone else in society. For Outer Party members like Winston Smith, life is incredibly hard. They're forced to endure chronic shortages of basic goods and live in dilapidated old housing where the walls are falling to bits and the heating regularly breaks down. Also, they are placed under constant surveillance by the notorious telescreens, which monitor their every movement.
Contrast that with the opulent lifestyle of the Inner Party: they get to live in swish, elegantly-furnished apartments, waited on by servants who cater to their every need. In what is supposed to be a radically egalitarian society, the Inner Party elite live like lords, a class apart from the rest of society. Their opulent lifestyles expose the utter hypocrisy on which their ideology is based.
For good measure, they also get to switch off their telescreens, albeit for only half an hour at a time. As only the most loyal and fanatical members of the Party get to be part of the elite, there's simply no point in subjecting them to the same degree of scrutiny as those in the Outer Party.

To answer this question, take a look at part 2, chapter 8. In this chapter, Winston and Julia go to O'Brien's apartment and they get a glimpse into the kind of privileges that he (an Inner Party member) has over them (Outer Party members).
Firstly, from this chapter, we see that Inner Party members live in a separate part of town. Here, their apartments are the epitome of luxury and not the dilapidated, run-down kinds of homes that the Outer Party members live in. O'Brien, for example, has soft carpets on the floor and elegant décor.
In addition, Inner Party members have a servant to make their domestic lives easier. Note that Outer Party members do not have such a privilege.
What we also find is that Inner Party members enjoy access to more varied food and pleasure items. O'Brien, for instance, enjoys fine wines and "good food."
Finally, and most importantly, Inner Party members can turn off their telescreens. This gives them a level of privacy and independence not shared by the Outer Party members.