In 2013, Williams finished ninth in the constructors' championship, with just five points to their name.
Last season was a different story, however, as the duo of Valtteri Bottas and Felipe Massa were fourth and seventh respectively in the drivers' standings, combining for a total of 320 points.
That was enough to see Williams finish third, 104 clear of fourth-placed Ferrari.
But speaking at the Autosport International Show on Thursday, Symonds believes taking the next step and challenging the likes of Mercedes and Red Bull is an altogether different proposition.
"We have set the bar high," he said.
"We are not a team that is in financial troubles by any means, but we are not a team that has the budget available that our competitors have.
"It is one of the things that gives me great pleasure in beating Ferrari last year, it is not the fact that I am a competitive person, but that we beat them on half their budgets!
"But moving on from third onwards is probably more difficult than it was moving from ninth to third.
"We can see that a lot of our rivals have either new power units or new people in place, so we have to exploit our advantage of the continuity we have got. We have got to push on."