Public Law ##-### is a reference to a slip law -- an actual bill, as passed by Congress and signed (or vetoed, if the veto was overridden) by the President. The first number is the number of the Congress that passed it, the second the number of the law in that Congress. (the "Public" is in contrast with private bills, which are things like "XYZ person, who is otherwise ineligible for citizenship, is a citizen" -- things that affect basically one person).
The US Statutes at Large are a compilation of slip laws (both public and private). Each volume has all the slip laws from a session of Congress, at least these days (I'm not sure how it interacts with the first few Congresses before the Statutes at Large existed). Laws there are still often called PL such-and-such, because that just means "law as enacted." If that doesn't line up exactly with the enrolled bill as passed and signed, something has gone wrong that really shouldn't go wrong. If this happens, someone is getting fired.
The slip law and Statutes at Large are both official, pretty much irrefutable evidence of any laws of the United States. Laws are passed by Congress, and they contain exactly what was passed. Logical. However, while they're logical, they're also a terrible research tool. If you want to find the law from them, you need to scan through every federal law ever passed. They aren't organized in any way having to do with topic. But there is another way: instead of just saying "everything the legislature has done is the law," you can rearrange those laws by topic and update them as the legislature does things. This is not easy: the legislature is passing things organized by what they're trying to do, and you need to put it all in an order that's based on what the laws actually regulate. There's a lot of editorial judgment involved. But it makes a better research tool to see what the law is.
The US Code is the second attempt at that (the first attempt failed). It's made by the House of Representatives Office of Law Revision Counsel, and does not inherently form part of US law. By default, it's merely strong evidence for what US law is; it is not conclusive, and the Statutes at Large takes precedence. This is because codification is hard. However, some titles of the US Code have gone further: the House OLRC cleaned them up and Congress enacted them into law. With the titles where this happened, Congress then passes all laws about them with direct reference to sections of the US Code, the OLRC can't move things around by themselves. The title itself becomes US law.
With these titles, the US Code is just like the Statutes at Large: it's identical to the law as passed, and if not then heads will roll. Also, in such cases, the US Code becomes just as official evidence as the Statutes at Large.
