If you use \\ in the middle of a paragraph, that means that the paragraph ends with an empty line. However, this empty line will make part of the paragraph, and is not broken at the start of a new page.
To break and get the space of an empty line, type
\vspace does not add lines, but space, and you don't want a line but space.
Remember that if you tell LaTeX to place an empty line, an empty line it will place. Even if that is at the beginning or at the end of a page.
And adding vertical space between paragraphs by using \\ is also not accepted by LaTeX ("Latex error. There's no line here to end.").