One key element is the new characters, especially Songbird. She plays a crucial role in driving the story forward. Another is the political intrigue within Night City. The power struggles between different groups add a lot of depth. Also, the dangerous missions that V undertakes are key. They expose V to new areas of the city and new threats.
The main story first. You'll understand the basic world and gameplay mechanics better. It sets the foundation for everything in the game. Then, when you move on to 'phantom liberty', you can fully appreciate the new elements as an addition to what you already know.
It depends on your preference. If you want to fully immerse yourself in the rich lore and world - building of the game right away, starting with the main story might be a good choice. It gives you a broad overview of the game's setting, characters, and main conflicts.
Since I don't know the specific 'phantom liberty story', I can't say for sure who the main characters are. But they could be people who are either fighting for or against this concept of phantom liberty.