When to use <span> instead <p>?

Semantically, you use <p> tags to indicate paragraphs. <span> is used to apply CSS style and/or class(es) to an arbitrary section of text and inline elements.

You should keep in mind, that HTML is intended to DESCRIBE the content it contains.

So, if you wish to convey a paragraph, then do so.

Your comparison isn't exactly right, though. The more direct comparison would be

When to use a <div> instead of a <p>?

as both are block level elements.

A <span> is inline, much like an anchor (<a>), <strong>, emphasis (<em>), etc., so bear in mind that by it's default nature in both html and in natural writing, that a paragraph will cause a break before and after itself, like a <div>.

Sometimes, when styling things — inline things — a <span> is great to give you something to "hook" the css to, but it is otherwise an empty tag devoid of semantic or stylistic meaning.

The <p> tag is a paragraph, and as such, it is a block element (as is, for instance, h1 and div), whereas span is an inline element (as, for instance, b and a)

Block elements by default create some whitespace above and below themselves, and nothing can be aligned next to them, unless you set a float attribute to them.

Inline elements deal with spans of text inside a paragraph. They typically have no margins, and as such, you cannot, for instance, set a width to it.

Span is completely non-semantic. It has no meaning, and serves merely as an element for cosmetic effects.

Paragraphs have semantic meaning - they tell a machine (like a browser or a screen reader) that the content they encapsulate is a block of text, and has the same meaning as a paragraph of text in a book.