Algebra I & II teach you the mechanics of equation manipulation, viewing functions in multiple formats, and creating some comfort with interpreting word problems.
Pre-Calculus is a bridge between what you learned in Algebra I & II and what you will learn in your four semester Calculus sequence (I, II, Multi, Diff-Eq). It certainly introduces the trigonometric functions and delves deeper into the transcendental functions, but along with it comes the ideas of limits and thinking about how a function is changing rather than just it's value at a given point.
When I went back to school I started off in Pre-Calculus to reinforce my foundations (I'm taking Multi now). Your algebraic foundations are incredibly important, and you shouldn't underestimate them. I had university credits for Calculus classes, but chose to start off again in Pre-Calculus. The biggest problems I see people have in Multi are with algebraic manipulation of large trigonometric functions, and working with multiple fractional exponents - Pre-Calculus stuff.
Developing the ability to "see" real-world phenomena as a mathematical function, and having the training to "tweak" the components of that function to model the data is a great skill that you will develop as you learn more and more mathematics. Pre-Calculus is the class where I first noticed this tool being forged.
Good luck with Pre-Calc! It's exciting to approach the Calculus sequence!