usepackage{tikz-cd} vs usetikzlibrary{cd}

No (practical) difference. The code (excluding copyright comment lines) in tikz-cd.sty is:

\ProvidesPackage{tikz-cd}[2014/10/30 v0.9e Commutative diagrams with tikz]
\RequirePackage{tikz}[2013/12/13] % pgf version 3.0.0 required
\usetikzlibrary{cd}

\endinput

so it just loads TikZ and then does \usetikzlibrary{cd}.

This is just a pedantic comment: there can be differences.

This is an admittedly nasty example.

This code works¹:

\documentclass{article}
\usepackage{tikz-cd}
\pgfmathdeclarefunction*{axis_height}{0}{\edef\pgfmathresult{1}}
\begin{document}
Hello
\end{document}

This code does not work:

\documentclass{article}
\usepackage{tikz}
\pgfmathdeclarefunction*{axis_height}{0}{\edef\pgfmathresult{1}}
\usetikzlibrary{cd}
\begin{document}
Hello
\end{document}

However, this works¹ again:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{cd}
\pgfmathdeclarefunction*{axis_height}{0}{\edef\pgfmathresult{1}}
\begin{document}
Hello
\end{document}

It is absolutely true that this is not a very useful code, and I have just designed the second code to crash. It does, however, show that there is a technical difference between loading tikz and then the cd library among other libraries (which could in principle do something along the lines of \pgfmathdeclarefunction*{axis_height}{0}{\edef\pgfmathresult{1}}), or later, and loading tikz-cd, which forces a certain ordering of loading of the libraries.

Of course, the cd library is, even though it is not part of pgf, a standard library and the authors of other libraries will most likely look at the cd code to make sure that they do not do harm to it, though it is not inconceivable that someone else may want to define function with the name axis_height. Nevertheless, if one is very strict, one has to say that it can, at least in principle, make a difference whether one load the tikz-cd package or the cd library. As I said, this is just an extended comment, but it does show that one cannot be absolutely sure that there is no difference.

_{¹ "Works" means you do not get an error when you compile the snippet. You should definitely not overwrite the function axis_height if you actually want to use the cd library.}

Additionally, in ConTeXt the tikz-cd module is not available, you need to do:

\usemodule[tikz]
\usetikzlibrary[cd]