Note that these are discussions about style, not something like grammar; as such, there is no "correct" or "incorrect" way among the different choices. The best thing is to pick one style and stick to it. And, APA and MLA are just two out of dozens of commonly used styles of citation.
When considering which style to follow, you should also consider the needs of the reader. APA and MLA are definitely common in many academic circles, and the citation method makes it really easy to cross-reference with other related work.
I've always been more comfortable with APA, mostly because it's what I had to use at school, but MLA advocates more precision. Whereas the APA style only mandates including a page number for direct quotations, if my memory serves correct, MLA requires page numbers even for paraphrased materials.
The number format to me always seems more relevant when the references themselves are less important. Compare the following:
- You're writing a research proposal, where you're including a literature review. Obviously, you want to show why your research is important, so you put your research in context of other studies that have been done before. In such a case, I would find the number format very inconvenient. You're referring to other studies, so the author-date format tells the reader immediately what study or researcher you're referring to. They don't need to flip to the end of your report and look up a number to figure out whose work you're referring to. The information you're referring to is specific to a study, and not something that might be, say, factual.
- You're writing a history paper and covering some famous historical event that many people have written about, and the perspectives don't necessarily differ widely. Still, you want to validate your information by providing sources. In this case, the author-date format might be a distraction. It might be more convenient for the reader to look up the source if they are interested in further detail about that little tidbit of information that you've cited.
Again, as I've mentioned, these are not rules but matters of style. Style is often selected with intent. So, my advice here would be to clarify your intent, and, based on that, decide which style to follow. (Unless, of course, you're forced to follow a style by your institution!)
I've seen, for instance, some books published with no indication in the text of cited materials, but at the end of the book, chapter by chapter, they will list the facts mentioned (in the order that they were mentioned) and the source or sources for each fact. This is the ultimate in distraction-free reading, but won't be very convenient for, say, a faculty or an editor to evaluate your work. Take a look at the following two images:
This is the source text, from page 20 of the book "The Big Necessity". Notice that there are no citations in the text at all.
But, this is not common knowledge. Where are the sources? The publishers decided to mention the page number, followed by the fact, and then the source. Here's the notes section for pages 18 to 22. You can see the facts about cell-phones and bras mentioned in italics, and the relevant sources.
You may also want to take a look at the Chicago Manual of Style which suggests several different ways of citation. You can view their "quick guide" online; on that page you'll see tabs for the notes and bibliography (number) format, and the author-date format. If you're serious about publishing, you should definitely have a copy of that book on your desk.