This is a potentially interesting question, now that Matthew Hoggard's Test career is almost certainly over. Raised principally because of the England 1995-2008 team thread (a period which nicely enough encompasses both bowlers' entire careers). Both of these two were bowlers short of the highest class, whose basic inherant skill level was not, quite, sufficient to give them the potential to achieve excellence, never mind brilliance. Both of times were left leading their country's attacks, a role you could tell they were certainly short of the class to do to a particularly accomplished degree. Yet both had disadvantages in their careers which meant that in the end they probably came out on the "moderate" rather than "good" platform. To an extent, to me, both go down as wasted talents. With less ill-fortune, I reckon both could've been very good, complimentary, bowlers.
Yet both had their moments aplenty at Test level and both left the impression that their side would've been weaker, at some points, for their non-presence.
Hoggard was useless between 2000 and 2003, but from 2004 to 2007 performed pretty well, taking advantage of what friendly conditions he encountered and twice putting-in some fabulous shows when the odds were stacked against him. But in the end his career was ended premuaturely and for most of his home career the ball used did not swing anywhere near as much as you'd expect a ball to do in England.
Cork in 1995 and 1996 managed some stirring, and to my mind quite excellent, bowling, and rehashed this in the summers of 1998 and 2000. Sadly, in the meantime, he did very little of note, but bad selection played its part, considerably so IMO, as did the unfortunate, and deeply untimely, breakdown of his marriage.
Before thinking the above through I'd be inclined to go for Cork, purely (possibly) because he was the first England bowler whose above-the-ordinary skill I really took note of. However, all things considered, perhaps Hoggard in the end achieved a little more. Certainly Cork never produced (even if he never got THAT many chances) one, never mind two, performances of the calibre of Nagpur 2005/06 or Adelaide 2006/07. But for those two, they're on an equal footing. However, I think that those two matches just tip things towards Hoggard.
Also, Hoggard ended-up playing for far longer for one simple reason - his temperament was better than Cork's. Cork should've been a fixture in home Tests (and most away ones, perhaps with the exception of those to the subcontinent) from '95 to '02 at the absolute earliest, if not a season or two later. That he wasn't was down to his inability to keep a level head. Hoggard excelled in this department, even if his powers of swing and accuracy were a little inferior to Cork's.
So, surprising myself, I think I'd have to vote Hoggard here on a head-only basis. Can't quite bring myself to do it though, so Cork it is.
And please - bowling only. Cork obviously added far more value with the bat than Hoggard did, but ignore this in this poll please.