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What is the difference between a Smarta and an Advaitin? Both are the followers of Adi Sankaracharya only.
People who worship Lord Shiva are called Shaivite and those who worship Lord Vishnu are called Vaishnavite. Then what is one called when he worships both Lord Shiva and Lord Vishnu only?
As I discuss in this answer, the word Smartha originally just meant "follower of Smriti. (Hinduism has two sets of scripture, Shruti or "that which is heard" and Smriti or "that which is remembered.) But the Smartha came to denote the followers of the sect founded by Adi Shankaracharya, which he did in order to create a more streamlined version of Hinduism. Let me explain.
In Adi Shankaracharya's time, Buddhism and Jainism had great influence in India. These two religions rejected the authority of the Vedas, and they attacked the Yagnas (fire-rituals) prescribed in the Vedas as cruel and barbaric. Unfortunately, the Hindu philosophers of the day were woefully unprepared to defend their religion, because the dominant school of Hindu philosophy at the time was Purva Mimamsa, a school which considered the Vedic hymns as eternal truths, but doubted the existence of the gods mentioned in those hymns! So most Hindu philosophers at the time were just as atheistic as the Buddhists and Jains they were debating.
In contrast to the Purva Mimamsa school, the Vedanta school of Hindu philosophy accepted the existence of the gods as well as the abstract entity Brahman. Adi Shankaracharya subscribed to a particular version of Vedanta called Advaita Vedanta, according to which the gods are just manifestations of Brahman, Brahman and the individual soul are identical, and the physical world is an illusion. With this belief system he was better-equipped to defend Hinduism than the followers of Purva Mimamsa.
But there was a problem: Hinduism as it existed at the time was in Adi Shankaracharya's view too ritualistic and disorganized due to diverse parochial beliefs and customs. So it was hard to defend Hinduism as it was actually practiced from Buddhist and Jain critiques. This motivated Adi Shankaracharya to create a more streamlined version of Hinduism. First of all, he tried to put an end to some traditional Hindu practices which he felt were justifiably criticized by non-Hindus (like animal sacrifice). Second of all, since he believed in Advaita Vedanta and thus believed that all the gods were just manifestations of a single Brahman, he saw no reason why anyone would need to worship so many different gods.
So he started to encourage people to worship just five gods, the Panchayatana: Shiva, Vishnu, Durga, Ganesha, and Surya. Other than Surya, the other gods in this list are ones that are mainly emphasized in Smriti texts like the Puranas, so the Adi Shankaracharya's new sect soon came to be called Smartha. Now you might have heard Smarthas simply being called Shaivites. There are two reasons for this. First of all, Advaitins tend to refer to the divine by the terms Ishwara and Parameshwara, which are often considered names of Shiva. Second of all, while Smarthas are theoretically supposed to worship five gods, usually the one they put the most emphasis on is Shiva.
To sum up, Advaita Vedanta is a (sub-) school of Hindu philosophy and and Smartha is sect of Hinduism, but in practice people who subscribe to one subscribe to the other as well.
P.S. You asked what to call someone who worships Vishnu and Shiva only. Well, there's no major sect if Hinduism that does precisely that. The closest you could come up with is follower of Harihara.
