The ancients were well aware that the planets behaved differently than the stars. Although the stars seem to travel across the sky, they always stay in the same position relative to each other. The planets do not do that. Not only do they progress across the sky independent of the stars, they also do so independent to each other. Clearly these five objects were outliers (the five planets visible without telescopes and thus known to the ancients are Mercury, Venus, Mars, Jupiter, and Saturn). Different cultures had different theories as to what the planets were.
The Egyptians, at least by later in their history, had a tendency to differentiate between the Sun and Moon (called the “two luminaries”) and the planets, which in one document are called the “five living stars”. The Egyptians associated Jupiter, Mars, and Saturn with different aspects of the god Horus. Venus was associated with Osiris, and Mercury with the evil god Seth. The Egyptians kept detailed observations of planetary motion, which were detailed enough for them to notices the little retrograde kink in planetary orbits that occurs due to their elliptical nature. They often left charts of star and planetary movements on the insides of coffins.
The Babylonians had recorded individual names for the five visible planets as early as 1800 BC, and viewed all five (plus the Sun and Moon) as manifestations of gods. Babylonian astrology was heavily linked with their religion, and the movements of the planets were seen as messages from the gods to humans.
The Greeks and Romans were heavily influenced by the cosmology of Plato and his student Aristotle, who refined it. Aristotle viewed the Earth as a sphere surrounded by other spheres which moved in a medium called the aether. Each sphere was the residence of either the Sun, Moon, stars, or one of the five planets. The independent rotation of each sphere accounted for the different motions of each of these bodies in the sky. The Greeks still didn’t view the planets as distinct worlds, and their term for them asteres planetai meant “wandering stars”. The Aristotelian model remained the dominant one in Europe until Copernicus.
In summary, the ancients were well aware that the planets were somehow different than the stars, and often gave special significance to them. There was no inclination among them that they were distant worlds, however.
Further reading:
The Planets in Ancient Egypt
Explaining Babylonian Astronomy