If you ever want to work tolerances out like this, then look at the max and min figures on the data sheets and see what the range is, then pick something inside that range. In this case, the Picaxe pin output "high" voltage (assuming a 5V supply) could range from close to the supply voltage (5V in this case) to 4.3 V, depending on the load. The Vf of the LED in the SSR varies from a min of 1.1 V to a max of 1.7 V.
Assuming the worst case to get at least 5 mA through the LED, this means taking the lowest Picaxe output voltage of 4.3V and the highest LED Vf of 1.7 V, which is a voltage difference of 4.3 - 1.7 = 2.6 V. With a 470 R resistor, the current flowing to the LED for this condition would be 2.6 / 470 = 5.53 mA, just over the 5 mA minimum required for the LED to operate (with a 680 R resistor the current drops to 3.82 mA, not enough to meet the LED specification).
Checking the worst case in the other direction, with the maximum Picaxe output voltage and the lowest LED Vf, gives 5 - 1.1 = 3.9 V. The maximum current flowing into the LED with a 470 R resistor would then be 3.9 / 470 = 8.3 mA, well within the 20 mA maximum allowable.
These two quick calcs show that 470 R is about optimum in terms of minimum power consumption and heat dissipation, combined with guaranteed operation of the SSR over the full range of the data sheet specifications.