Are the rules in the Player's Handbook different from the Starter Set rules or just more expanded? Can you carry your level 5 character created from the Starter Set forward into "regular" play?
It's all the same game
It was part of the design of 5e D&D that there should be different entry points to the one game. These ended up being:
- a boxed set of "board game" complexity (the Starter Set)
- a free version of the core rules with few options (the Basic Rules)
- the full version of the game, with lots of options (The Player's Handbook, Dungeon Master's Guide and Monster Manual)
But they are all the same game, and all completely compatible, and you can continue using your Phandelver characters with the full ruleset.