Statistical methods vary according to the measurement levels of variables, and it can make a difference if variables are classified as nominal, ordinal, or interval/ratio. But I've never seen any statistical test that depends on the interval/ratio distinction.
Leaving aside for the moment any philosophical objections to the "standard" classification of measurement levels, is there any statistical reason to care about the difference between interval and ratio measurement levels?
As an example, consider a variety of generalized linear models -- for example, gamma, Poisson or inverse Gaussian regression models. Those models (plus some others) correspond to models for data that could be classified as ratio data (the zero is meaningful, 2 really is twice as much as 1, etc).
Further, these models are not equivariant to adding or subtracting constants to/from the data (so it's plainly not simply interval).
If you're trying to analyze some kinds of times/counts/monetary data -- and numerous other kinds of data, such models may be very useful.
In a related fashion, a log-transform or a power transform isn't generally meaningful for data with an arbitrary zero (where the data mean exactly the same thing with a different zero) even if the data happen to be all-positive. It is often meaningful for ratio data.
One should also be careful not to get too hung up on Stephens' typology in particular; there are other typologies that can be more useful for statistical purposes; indeed the notion of using typologies to decide analysis (rather than the question the data are being used to investigate or answer) may often be misplaced. It can sometimes be helpful, but not infrequently, it's the opposite.
Computing percentages or ratios seems to be the easiest statistical operation that is valid for ratio scale but invalid for an interval scale.
Temperature in Celsius or Fahrenheit are examples of interval scale measurement (no fixed 0). For example, it makes no sense to say that 110°C is 10% hotter than 100°C, because if you transform the temperatures to Fahrenheit you would get $\frac{230-212}{212}\times100\%=8\%$ hotter.
However length has a fixed 0, and is on a ratio scale. So now, you can say that something measuring 110 cm is 10% longer than something measuring 100 cm. If you transform that to inches you still get $\frac{(43.3071 - 39.3701)}{39.3701}\times100\%=10\%$
