For most celestial bodies, there's no difference. You tell it to go into orbit, and it just stops there. The icon on the sidebar will be red, showing it as idle.
For some bodies, specifically planets that you own, it will put the ship in orbit, which is different than simply being nearby. The icon on the sidebar will be green, indicating orbit.
When in orbit around one of your planets, fleets:
- have reduced upkeep costs
- can be repaired
- can be upgraded
- can be affected by spaceport modules, such as the Crew Quarters which reduces upkeep costs
- will merge with any new ships constructed at the planet
Fleets also need to be in orbit in order to bombard a planet, and transport fleets must be in orbit in order to land their troops.