@JC, I may be missing something, but from my "googling" it appears that the National Express bus service to Stonehenge is only to the town of Amesbury which leaves you fairly distant from the visitor center. According to http://www.stonehenge.co.uk/arrive.php "Get off at Amesbury. From there you can either walk (about 2 miles) or get a taxi." Does that sound correct?
Yes, the National Express doesn't go the Stonehenge visitor centre, it goes to Amesbury which is a gentle 2 mile stroll through some interesting countryside, Stonehenge isn't all about the stones, there are the neolithic and bronze aged burial mounds surrounding it and many other aspects intrinsic to the creation and believed purpose of the stones which you'll be passing through. Of course you could always get a taxi.
The major point however is the usefulness of the English Heritage pass and if used well will more than pay back its cost. Will the National Express Coach and additional taxi outweigh the cost of the Salisbury Red bus? Who knows but you're visiting some of the most interesting parts of a country in the world and your decision making rests solely on the difference of £tens. Throw caution to the wind, buy the English Heritage pass, take the direct bus to the stones and forget the relatively small additional expenditure, you only live once!
Oh and whilst you're in the area be sure to visit the stone circle in Avebury. It's different to Stonehenge, is on a bigger scale and you can walk amongst and touch the stones, If I had to choose I would say I prefer Avebury over Stonehenge. Just don't eat at the Red Lion pub within the stone circle, the food is crap and the packs of frozen, brought in food the chef collected from one of the outbuildings didn't inspire confidence that it was all hearty, home cooked fare.