The difference between rationalism and empiricism is in the method of proof.
Empiricism requires, by the conventional definition and usual understanding of the term, that the proof of your claim be an observation of the material world.
Rationalism requires instead that the proof of your claim be an observation of your own mind.
The widespread contention that this distinction is fundamental is spurious, however. Empirical sciences wouldn't exist as we know them had they to be justified only by observations of the material world. Empirical sciences require from scientists that they first observe their own mind since the percepts they have relative to the material world are all entirely mental events.
And of course you only need to read Descartes's first pages leading to the Cogito to be convinced that understanding the idea of it requires understanding his confrontation of his observation of his own mind, through introspection, and his observation of the material world, through his senses (irrespective of whether any of these things exist as such).
The difference is real but more a matter a degree than of a black-and-white distinction. Science relies on a large extent of what scientists themselves call "thought experiment".
One of the first scientific discovery, and one which is well-known the world over, is Archimedes' principle. The principle states that water exerts an upward force upon any body partially or fully immersed in it and that this force is equal to the weight of the water displaced by the body. Archimedes didn't discover this principle as it is formulated now, but he realised how he could prove whether a crown is of impure gold by immersing it into water. His idea was to measure the volume of the water displaced by the crown as a measure of the volume of the crown. Archimedes is supposed to have shouted "Eureka" while having a bath and presumably observing the level of the water inside the bathtub go up as a result of immersing himself. However, anyone having a bath can experience the Archimedes force as exerted on their own body by water. Swimming certainly seems a lot easier for it. We feel the force. It is an empirical fact. Yet, understanding what is the cause of the force requires an operation of the mind, something entirely rationalistic in essence.
Still, the ultimate proof in the empirical sciences can only be an observation of the material world. If physics, the epitome of empirical sciences, evolved to the point where such a proof could no longer be obtained, it would be perceived more as a rationalist discipline, as seems indeed to be the current status of String Theory.
It is also possible to consider our observation of our own mind (introspection) to be fundamentally an empirical activity.
There is indeed no fundamental difference between observing the material world and observing the pain that you experience whenever you experience pain. The difference is entirely in the fact that two observers will be able to agree that they see a tree or a bird, while only one observer will be able to observe the pain experienced. Yet, we all will experience pain at some point in our lives, as well as such mental phenomena as remembering the past, feeling nauseous, having a logical intuition, having a Eureka moment, etc. And indeed having any idea about the empirical world.