For both linear and nonlinear regression of replicate values, Prism gives a choice of fitting individual replicates vs. fitting the means. What's the difference? Which should I choose.

If you enter individual replicate values, you can specify (in the Weights tab) whether Prism should fit the individual replicates or fit the means.  By default, Prism fits each replicate as an individual value.

Which approach to choose? 

Choose the approach that matches your experimental design.

You should consider each replicate a separate point when the replicates are independent. Two examples:

• You are doing a radioligand binding experiment. All the data were obtained from one tissue preparation and each replicate was determined from a separate incubation (separate test tube). The sources of experimental error are the same for each tube. If one value happens to be a bit high, there is no reason to expect the other replicates to be high as well. The errors are independent.
• You are doing an electrophysiology study. You apply a voltage across a cell membrane and measure conductance. Each data point was obtained from a separate cell. The possible sources of experimental error are independent for each cell. If one cell happens to have a high conductance, there is no reason to expect the replicate cells (those that you apply the same voltage to) to also have high conductance.

Average the replicates and treat the mean as a single value when the replicates are not independent. Two examples:

• You performed a binding experiment with a single tube at each concentration, but assessed the radioactivity in each tube three times. Those three values are not independent. Any experimental error while conducting the experiment would affect all the replicates.
• You performed a dose-response experiment, using a different animal at each dose with triplicate measurements. The three measurements are not independent. If one animal happens to respond more than the others, that will affect all the replicates. The replicates are not independent.

How different will the results be?

If you fit only the means, the R² value will be much higher. Think about it this way. If the line or curve goes right through the means, the R² will be 1.0 if you fit the means. But you fit the individual replicates, it is impossible for R2 to ever be 1.00 if the replicate differ from one another. Fitting individual replicates increases the variation that Prism sees, so lowers R². 

If you fit only the means, the sum-of-square, of course, will be much smaller. You are fitting fewer points, and these points (means) will be closer to the curve on average. So the sum-of-squares will be much lower. 

Parameters and confidence intervals? The values of the parameters, and their standard errors and confidence intervals, will vary between the two methods, but usually not by very much, and not in a predicable direction.