When Prism compares the fits of two models using AICc, what determines if the delta AICc is positive or negative?

Prism compares two models, one simpler and one more complex. It doesn't matter which model you pick on the Fit tab, and which you pick on the Compare tab. The simpler model is the one with fewer parameters to fit, so more degrees of freedom. 

Prism reports the actual (not absolute) value of the difference in AIC between simple model and complex model in the result of model comparison, therefore the value of the difference can be negative. It calculates the difference as the AICc of the simpler model minus the AICc of the more complicated model. When the more complicated (more parameters) model has the lower AICc and so is preferred, Prism reports the difference of AICc as a positive number. When the simpler model has the lower AICc and so is preferred, Prism reports the difference of AICc as a negative number. 

Prism reports which model is preferred, and the relative likelihood of each model. There really is no reason to look at the AICc values, or their difference, except to compare with other programs or text book examples.  

Note that the actual value of the AICc is irrelevant. If you change the units of your Y values, the AICc will change. It might even change sign, between positive and negative. All that matters is the difference between the AICc values. Take the difference between the two values, not the absolute values of the AICc.