Danny Haynes had given Ipswich the lead with a fierce low shot on 19 minutes, but Fuller, who spent time on loan at Ipswich two years ago, scored with a similar effort 14 minutes later.
Both sides had chances to claim all three points, with Fuller perhaps having the best when he broke through 16 minutes from time but was unable to round home keeper Neil Alexander.
Both teams got into their stride early with Alexander saving a low Fuller shot and then Alan Lee played in Jon Walters, who shot across goal on his weaker left foot.
Steve Simonsen stretched to turn away a shot-cum-cross from the livewire Haynes and Lee almost got his head to a Jason De Vos chip in the penalty box as Town tried to turn the screw early.
Danny Pugh volleyed over from an Andy Wilkinson cross, before Alexander had to dive to turn aside a 20-yard low drive from Richard Cresswell as Stoke got a foothold in the game.
Town went in front in their next attack when Lee's ball wide was collected by Haynes ahead of Dickinson and he drilled a shot across goal and in off the far post.
Ipswich could have built up a comfortable lead in the space of four minutes as first Haynes had a fierce drive parried away by Simonsen, Ryan Shawcross cleared off the line from Walters and then Lee glanced a Tommy Miller corner wide.
Shawcross and Leon Cort got in each other's way when either could have scored from a Stoke corner, but the visitors then levelled when Pugh pulled down a long ball and his pass was collected by Fuller, who drove across Alexander from a narrow angle.
The second half took a while to get going with Lee side-footing over a David Wright cross, then for Stoke Pugh had a shot held low down by Alexander and John Eustace fired well wide from a loose clearance.
The game became a bit fractious and Fuller's reaction to several De Vos fouls eventually got the home skipper booked, before Fuller was cautioned himself for needlessly charging into Alex Bruce to the delight of the home crowd.
Haynes had a 25-yard free-kick deflected over, but Stoke nearly scored from the ensuing corner as it was cleared up field to Fuller who raced away from Wright and Sito Castro, but as he tried to round Alexander the Scottish international dived onto the ball.
Ipswich looked set to take the points on 82 minutes when Haynes got free down the right and crossed low for Lee, but what looked like a tap-in was denied by a superb block by sub Gaby Zakuani.
Both sides had penalty shouts turned down late on as Pablo Counago felt he was pulled down at a corner and then Stoke sub Jon Parkin went to ground as Alexander challenged him, but referee Keith Stroud turned both shouts down, and a draw was a fair result in the end.