[keep in mind while reading the following that it has been more than twenty years since I read these plays. If you've read either author in this century your opinion is probably worth more than mine.]
I know there are some who would dispute this but I think most critics and literary scholars would, with all due respect to both writers, rank Beckett over Pinter. It's not a position I'm prepared to defend (though I am inclined to agree with it), but, if you are willing to go along with it, the proposition suggests a somewhat counter-intuitive result: the greater writer is the less influential.
I'm using influence here in a fairly specific way to mean later writers' use of specific elements associated with Beckett's and Pinter's work, not to mean general impact. Lots of writers might tell you that see Godot changed their life without being able to point to anything specific in the play that changed in their work.
It's not difficult to why much of Beckett's best known work isn't that widely imitated. When you've written a play where the staging consists of three actors' heads sticking out of urns with the final stage direction "repeat play" and you actually name it Play, you haven't left future generations a lot to build on. I remember being more impressed by Play than by the Birthday Party, but I had no urge to go out and write my own actors-in-furniture cycle.
Lots of writers did go out and try to write their own versions of Pinter plays and many more have incorporated Pinteresque elements, creating a sense of absurdity or menace by the strangely off-kilter use of mundane and colloquial language. You can see Pinter's influence even in popular genres like science fiction (it was actually an arc of the no-budget British time-travel series Sapphire and Steel that got me thinking about this), horror (check out the bar scene in the Shining) and even sitcom. By Beckett's pop culture presence is largely limited to allusion (an unpunctual character might be compared to Godot, for example)
If you've played along up until now, I have a question for you: in your field, who are the Becketts and who are the Pinters?
For example, in mathematics I think you could make a case for Newton as a Beckett and Leibniz (due partly to his superior notation) as a Pinter. When I asked a musician friend, he immediately suggested Thelonius Monk as a Beckett, brilliant but something of a terminal point.
Other than that the competition is open. How would you complete the sentence: Though __________ was more brilliant, _________ was more influential.