chapotraphouse

The more noticeable cause of the sky looking different for the ancient Greeks would be due to precession instead of Earth's orbit around the Galaxy. Precession is Earth's "wobble", the "rotation" of Earth's own axis of rotation. Like how a top wobbles around as it spins. It takes about 26,000 years for the Earth's axis of rotation to make "wobble around" in one cycle. So this is the larger cause of the night sky, and the pole star, looking different for the ancient greeks. But this impacts the apparent position of all stars in the sky. So Ancient Greeks could see certain constellations that are currently too far below the horizon for their contemporaries. The positions of these constellations have changed.

Earth's or the solar system's orbit around the galaxy takes about 230 million years, so this would have less of an impact.

But there would be some differences.

The stars are moving though as they orbit around the Milky way. Some stars move much fast than others and their individual positions could definitely change over thousands of years. From Universe Today

When a star is moving sideways across the sky, astronomers call this “proper motion”. The speed a star moves is typically about 0.1 arc second per year. This is almost imperceptible, but over the course of 2000 years, for example, a typical star would have moved across the sky by about half a degree, or the width of the Moon in the sky.