Claim Let $f:\widetilde X\to X$ be a projective birational morphism between smooth quasi-projective varieties over an algebraically closed field. Suppose that the image $Y$ of the exceptional set of $f$ in $X$ is also smooth. Then $f$ is the blow up of $X$ along $Y$.
Proof By the assumptions $f$ is the blow up of $X$ along an ideal sheaf $\mathscr I\subset \mathscr O_X$. Let $\mathscr J$ be the ideal sheaf of $Y$ and let $\pi: B\to X$ be the blow up of $X$ along $Y$ (i.e., along $\mathscr J$).
Since by assumption the support of $\mathscr O_X/\mathscr I$ is $Y$, it follows that $\mathscr J$ is the radical of $\mathscr I$ or in other words $Y$ is the reduced scheme of the scheme along which one has to blow up $X$ to get $f$. This implies that $\pi$ factors through $f$: $$\pi: B\overset{\sigma}\to \widetilde X\overset{f}\to X,$$ i.e., $\pi=f\circ\sigma$ for an appropriate $\sigma:B\to\widetilde X$.
We may assume that $Y$ is connected and then if it is smooth it has to be irreducible. From the standard facts about blow ups along smooth varieties it follows that the exceptional set of $\pi$ is a single irreducible divisor on $B$ (it is a projective space bundle over the irreducible $Y$).
Now observe that if $\widetilde X$ is smooth (actually already if it is normal and has $\mathbb Q$-factorial singularities), then $\sigma$ is either an isomorphism or it has an exceptional divisor. The same holds for $f$, but there is only one divisor that is exceptional for $\pi$. Therefore either $\sigma$ or $f$ has to be an isomorphism. $\square$
Remark the proof suggests that even assuming that $\widetilde X$ is normal and has $\mathbb Q$-factorial singularities is enough for this characterization. Accordingly in the example below, when the ideal $(x^2,y^2)$ is blown up on the plane, then $\widetilde X$ is not normal and the blow up of the reduced point is the normalization (which happens to be smooth) of the pinch point surface $x^2z=y^2$.
Example (property 1 fails, but property 2 is satisfied)
Look for $f$ as the blow up of an ideal sheaf $\mathscr I$, so $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subscheme $Z\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Z=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Z$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Z\simeq \mathbb P(\mathscr{I/I^2})$.
Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.
Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.
To round things up Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth.