The first is not an integral; it's a differential. The second is an integral.
When doing substitution or integration by parts, one considers differentials. For example, to do integration by parts on
$$\int x\sin x\,dx$$
one can say "let $u=x$ and $dv = \sin x\,dx$." Then you want to find a function whose differential is $dv$, so you are trying to find $\int dv = \int \sin x\,dx$; we usually don't actually write this, and simply write "then $v=-\cos x$", which may be what is confusing you and leading you to believe that "$\sin x\,dx$" is some kind of integral. But it's not an integral.
I'm just guessing, mind you, since you have not yet provided explicit examples of the use you believe refers to integrals.