Some of the answers to this question — indeed the question itself — seem to assume that “genius” and “intellectual” have clearly contrasting meanings, such that an intellectual cannot be a genius, and a genius cannot be an intellectual. This is simply not the case; it may be fun and clever to think this way, but it is deeply unphilosophical and unphilological — inattentive to the nuances of language and the history of ideas.
In truth, “genius” and “intellectual” belong to different discursive regimes. The term “genius” traces back to Greek antiquity: it means a “demon,” a spiritual being, that accompanies and guides one. During the modern period, especially the 18th century, this transmogrified into the idea of an innate capacity for creativity and originality that, not bound by conventions or recognizable rules, seems to be wondrous, divine, and even a bit terrifying. Already for the Greeks, human skill was seen as something wondrous and frightening. (Compare the Greek deinos) The primary manifestation of genius was in art, and especially music, though it also was carried over to science and philosophy. (Schelling remarks, rather oddly, that Kepler was a genius, but Newton was not! ) But “world historical individuals “ like Napoleon were also geniuses of a sort.
During the 20th Century, due to the growing prestige of science and technology, the primary locus of genius was found in math and natural science rather than art. And at the same time, due to the development of psychometrics, genius came to be understood as an innate genetic endowment, characterized by unique computational powers. Hence the computer-like Von Neumann, architect of modern computing, became the exemplary genius.
The term intellectual in the modern sense came into use in the early 20th century. It has nothing to do with innate mental powers, originality, or creativity — though these things may be what distinguishes a “true intellectual” from a second-rate intellectual. An intellectual refers, fundamentally, to someone who is “at home” in the world of ideas that exist at the threshold between specialized and popular understanding; between the esoteric and esoteric. Scholars, scientists, philosophers, mathematicians are each dedicated to esoteric pursuits. You can be a scholar, scientist, philosopher or mathematician, or even an artist, without being at the same time an intellectual. You become an intellectual when you seek to engage with the “public” significance of these pursuits. In this sense, the concept of intellectual is closely bound up with the concept of ideology; ideas as unifying systems of belief that saturate the public sphere.
You can be an intellectual without actively engaging in the public sphere. But you have to be interested not just in scientific, philosophical, scholarly or artistic pursuits, but in how they interact with each other, and have a meaning, as ideas, that reaches far beyond the proper sphere of their understanding.
Marxism, psychoanalysis, existentialism, libertarianism — indeed, every “ism” — are the stock in trade of the intellectual because they are “inherently” ideological.
A scientist may be an intellectual or may not. Frank Wilczek is a brilliant physicist, but he is also an intellectual, insofar as, in his popular writings, he proves himself confessing in a wide range of subjects. Other physicists, however, have a much more narrowly circumscribed sphere of interests.
When intellectuals have no real grounding in any esoteric discursive regime, they are called pseudo-intellectuals. Yet, in a way, all intellectualism is pseudo-intellectualism, since no esoteric discourse can enter into the public domain entirely on its own terms. Vulgarization is inevitable.
Intellectuals may be further classified as: popularizers. pundits, ideologues, enthusiasts, and cynics.