Antiderivative vs. Indefinite integral

I spent some time today getting ready for my class for the next term.  In particular, I was reading through the sections on antiderivatives and indefinite integrals.  I had normally taken these things to be distinct concepts.  Given a function $f$, an antiderivative of $f$ is any function whose derivative is $f$.  The indefinite integral is the set of all antiderivatives.  Since there are infinitely many antiderivatives, while an antiderivative is a single function related to $f$ by differentiation, these are not the same thing.

Or at least, that's what I thought until I read the aforementioned sections. The definitions I read today did not seem to make that distinction.  I consulted with the textbook that I used in my own first year calculus class many years ago.  This textbook does make the distinction.

I suspect the difference between the two ideas is not appreciated by a lot of students.  We don't dwell on it much once we give the definitions.  After that, we usually teach them techniques for finding indefinite integrals, and as long as they remember to include the constant of integration, we're happy.

The textbook that doesn't distinguish the two is one of the most widely used introductory calculus texts.  It's quite an old version, though.  Perhaps newer versions do distinguish.

In any case, it made me wonder how important, if at all, the distinction is.  Was a serious error made by leaving it out?  Am I being too picky by insisting my students understand the difference?

One thing I do find somewhat bothersome about erasing the distinction is the abuse of notation that results.  If $F$ is an antiderivative of $f$, then it can be written
$$\int f(x) dx = F(x)$$
The constant of integration is omitted because there is no distinction between the two terms.  This is an abuse of notation, because there are actually infinitely many functions that could appear on the right, all differing form $F$ by a constant.  So the $=$ sign doesn't belong there, because it implies uniqueness.  Of course, even writing the constant of integration
$$\int f(x) dx = F(x)+C$$
can be seen as an abuse of notation, but at least it captures the idea that there are infinitely many antiderivatives of $f$ and each of them has the form on the right.  I suppose, to capture the essence of the idea, we could instead write
$$\int f(x) dx = \{F(x)|F'(x)=f(x)\}=\{G(x)+C|C\in{\mathbb R}\}$$
where $G$ is a known antiderivative of $f$, but I've never seen it written this way.