First of all, no indefinite article with proper names -
... there is very little difference between results obtained with and without Rashba coupling.
Here and means that you have two types of results, one obtained with the coupling, the other - without it.
if you say
... there is very little difference between results obtained with or without Rashba coupling.
it would mean that all the results were obtained the same way, either considering the coupling, or not.
But there is a bigger problem with the whole sentence. First of all, Rashba coupling, or Rashba effect, is a physical phenomenon, and you cannot obtain something with that effect, you can obtain something taking into account/considering that effect. Or otherwise you can say
Using the expression/equation for Rashba effect ...
Finally, I would stick to the following sentence to express what you meant to:
The consideration of Rashba effect introduces a very small difference between the results.
or, if you still want to go with your initial structure:
There is a very small difference between the results obtained with and without consideration of Rashba coupling.
Note again, that Rashba coupling cannot change your results, it is a model/phenomenon that always exists, the only difference is that you either take it into account or do not.