Let $X\subset\mathbb{P}^n$ be a smooth irreducible variety (over $\mathbb{C}$) and $p\in\mathbb{P}^n$ a point. Let $\pi:\mathbb{P}^n\setminus\{p\}\to\mathbb{P}^{n-1}$ be the linear projection with center $p$ and denote by $Y$ the Zariski closure of $\pi(X\setminus\{p\})$. I would like to know:
When is $Y$ smooth and the restriction $X\setminus\{p\}\to\pi(X\setminus\{p\})$ of $\pi$ an isomorphism?
If $p\not\in X$, then I think this is equivalent to $p$ not lying on the secant variety of $X$. But I am mainly interested in the case when $p\in X$. In this case we can rephrase the question as:
When is $Y$ the blow-up of $X$ at $p$?
The exact statement for surfaces is given in Beauville's book: Complex Algebraic Surfaces. Lemma 4.4. I imagine that this generalises easily to higher dimensions.
The condition is different whether or not the point is contained in the variety (as you mentioned), I will state both cases.
I quote exactly from the book, to avoid messing it up:
Lemma: Let S be a surface in $\mathbb{P}^N$ and $p \notin S$ (respectively $p \in S$). Let $f: S \rightarrow \mathbb{P}^{N-1}$ (respectivly $f: \hat{S} \rightarrow \mathbb{P}^{N-1}$) be the restriction of the projection away from $p$. Then $f$ is an embedding $\iff$ there is no line through $p$ meeting $S$ in at least $2$ (respectively at least $3$) points, counted with multiplicitly.
A nice corollary of this statement (given immediately after in Beauville's book) is that every smooth projective surface is isomorphic to a surface in $\mathbb{CP}^5$.