Rationalism tends to rely on deductively certain methods to establish the truth. Empirical truths are probable to a degree. It is a rational fact that given the theory of numbers, any number can be composed of a product of unique primes raised to a power. But is an empirical method only that can determine which prime number was used to generate a public encryption key. Rationalism can answer questions about logically certain relations, while empiricism cannot. Of course, the two methods complement each other as in math and science.
Conifold is right to characterize the difference between rationalism and empiricism as a question about the source of knowledge, which is an epistemological matter. This is strongly linked to the philosophical concepts of a priori and a posteriori. To get a good sense of the two forms of reasoning, one has to appreciate the role and characteristic of reason within the nature of theories of truth. So let's examine epistemology from the perspective of Robert Audi, who is an authority on the matter, and is cited heavily in the SEP article on epistemology. His introductory text on the matter is a bit of a tough read, but well worth it if you're looking to sharpen your metaphysical skills.
Audi recognizes five sources of knowledge: perception, memory, consciousness, reason, and testimony. Let's characterize rationalism broadly as the use of consciousness and reason, and empiricism as the use of perception and testimony. The former are innate in the sense that they are private and seem to develop somewhat consistently among thinkers, and the latter are ways to access our public lives; thus we have the foundations of subjectivity and objectivity. When Descartes in Meditations used his awareness of his own thinking as an axiom of reasoning from first principles, this is rationalism, and indeed, Descartes is known as a rationalist. This is very different thinking of relying on epistemological sources such as the perception of the external world, which is an empirical matter that would have been championed by Bacon, Locke, or Hume.
So, a rationalist tends to rely on reason or logic, and an empiricist tends to rely on experimentation or trial and error. Obviously, both are necessary aspects of good thinking if one is interested in connecting one's private world to one's public world. These two traditions in the 21st century have largely developed into an alliance between mathematical and scientific theories across a broad range of disciplines.
What can rationalism accomplish "prove" that empiricism cannot? That's an excellent question! But as in all matters philosophical, it depends on the turn of language. What is 'proof' (note well I have transitioned into use/mention distinction) is a matter of philosophical controversy and metaphysical presumption. Let's characterize the nature of proof in the two schools of thought: on the one hand, rational proof tends to be seen as consistency in propositions and might be seen as aligned with the coherency theory. Empiricism is the natural ally of the correspondence theory. This then hints at what rationalism accomplishes, that empiricism cannot.
Empiricism and induction are ways of using perception and testimony to build an understanding of the world by accessing the world. They are the things that we see, touch, hear, feel, or likewise accept as true from our communities. If you hear from many people that Socrates is in the kitchen of his house, and you go there and verify it for yourself, you have embraced truth from the fundamentals of empiricism, and you can participate in the community that agrees that's where Socrates is at the moment.
But if you were asked, is Socrates in his house? You simply needn't check for that if you take it that Socrates is in his kitchen. By deduction which is a rational method, you can demonstrate clearly that if he is in the kitchen, and the kitchen is in the house, then Socrates is also in the house. And that second conclusion requires no empirical evidence other than the first proposition and the fact of spatial transitivity. The second statement is true by deduction. Volumes that contain volumes that themselves contain volumes, actually contain both volumes (and so on). Matroyshka are a perfect toy to teach such a principle.
This becomes more obvious when we cannot use our senses first hand to observe something. Here's an exercise for you. Take a ream of paper, and measure it. You can guestimate if you'd like. Perhaps you take the number 5 cms (about 2 inches). Now, given a ruler, you can find a relatively reliable answer for the ream, but not for a single sheet. Even a coarse micrometer might not help. And yet, if you take 5 cms and divide it by 500 sheets, then you arrive at a conclusion. A sheet of paper is (on average) 1/10th of a mm (or if you prefer 100 micrometers). Note, that this is a deductive fact that relies on mathematical axioms. It cannot be observed with the naked eye. Empiricism can provide some of the inputs to the argument, but the argument itself relies on rational methods. And this is the answer to your question.