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My FRAP curve best bits a double exponential fit, but I get two tau values by using a double exponential fit, one small and one large. My question is which tau value do I use? Only the small tau value make sense in my experiments, am I allowed just simply to ignore the large tau value?

The double exponential fits to almost any FRAP curve (even for diffusion only recovery). This causes problem in interpretation of data: it could be that kinetics beihing is nothing to do with the double exponential.

If you know pretty well that the kinetics you are analyzing is indeed a double exponential reaction and doing the experiment correctly, you could interpret that there are reactions with fast and slow kinetics and could use those values to discuss differences.

If such condition is not met, forget about the model behind the recovery. Then all you could do is to compare the half-max time (1/2 tau). This is rather qualitative, but still is valid and scientific.