All else being equal, a thicker string will damp out transverse vibrations more rapidly because it experiences more drag (inter-molecular deformation) per unit length. (See section 4.6 of [not my work] http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf.) (If we consider strings made of different materials or under different tensions, this rule may or may not apply.)
As noted in some other answers (for various correct and incorrect reasons), the spectrum of harmonics will also be different between the two strings. However, while this is true, my own experience is that the difference in tone produced by this effect is usually smaller than what can be achieved with other performance techniques. For example, by moving the bow closer to the bridge when playing on the heavier string, the difference between the overtone spectra may become negligible or even inverted, or acquire totally different characteristics like the noise of attacking each string from an angle. Thus, getting your sequencer to sound like a human musician may be somewhat counterintuitive, and you should play around with different ideas.
In addition to adjusting the rate of decay of one of your enharmonic notes, I also suggest you try adding vibrato (finger-speed frequency fluctuations) to one of them, since that is what a real player's finger would be doing on the fingered string, but obviously not on the open string.
As far as shifting the center of each pitch, which is also a realistic idea, you would simply need to scale the pitch difference logarithmically with frequency, as @BenKushigian suggests. I also like @smiley's idea of adding a very slight phasor over the entire waveform, to enhance the effect of beating between the strings.
Please let us know what you find sounds best!