Im currently taking Pre Calculus and took Algebra 2 last year. To be honest, there really is no difference, there are only three new chapters of content we didnt go over last year in the whole book. If you passed Algebra 2 you'll pass Pre Calculus.
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Calculus is usually taught two years after Algebra two. Between Algebra two and Calculus is Trigonometry or Pre-Calculus. We teach it in A Texas school at grade 12.
Pre-calculus is supposed to be a stringent review of trig and algebra in preparation for calculus. So, pre-calculus, I would say.
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Simple answer: Calculus involves derivation and integration, precal doesn't. Pre calculus gives you some of the algebraic, geometric and trigonometric understanding that is required to comprehend the concepts in calculus. Without the knowledge from precal, calculus would not be easily understood, as it is taught in schools today.
You must have a strong basis in Algebra, Algebra II, Geometry and Trigonometry. Usually high schools offer a pre-Calculus course which is somewhat of a conglomeration of the aforementioned courses. Then you would move into differential calculus, integral calculus, vector (multi-variable) calculus, and finally differential equations, which is considered to be at the top of the hierarchy of the calculus courses. So take Algebra, Algebra II, Geometry and Trigonometry to get your strong foundation before begining the calculus sequence.