Is a von Neumann algebra just a C*-algebra which is generated by its projections?
No. Consider the commutative C* algebra $c$ of convergent sequences of complex numbers (with pointwise multiplication), corresponding to $C(X)$ where $X$ consists of a convergent sequence and its limit. It is also the norm closure of the span of its projections (which are the sequences of $0$'s and $1$'s that are eventually $0$ or $1$), but it is not a von Neumann algebra: the weak closure is $\ell^\infty$.
More generally, IIRC, $C(X)$ is the norm closure of the span of its projections iff $X$ is totally disconnected, but in order for it to be a von Neumann algebra $X$ must be extremally disconnected.
Approximately finite dimensional C*-algebras are generated (as Banach spaces) by their projections, but they are not von Neumann algebras except in the finite dimensional case. $c$ is an example of this, and the algebra of compact operators on a separable Hilbert space is another.
Blackadar's survey article "Projections in C*-algebras" contains a section called "Existence of projection properties" (starting on page 138), where this property is listed as one among many axioms that are compared ("LP"). It is mentioned there that in the case of $C(X)$ with $X$ compact Hausdorff, most of the axioms are equivalent to total disconnectedness of $X$, which is also equivalent to $C(X)$ being approximately finite dimensional if $X$ is metrizable.
As Yemon mentioned, work of Kaplansky on AW*-algebras is relevant. Piecing together a bit from Blackadar's exposition, citing work of Kaplansky and Kadison (and possibly Sakai, see 6.3.1 and 6.3.3), the following three properties (all at once, not separately) characterize a C*-algebra $A$ as a W*-algebra:
Every maximal commutative C*-subalgebra of $A$ is generated by projections.
The projections of $A$ form a complete lattice.
$A$ has a separating family of normal states.
Adding to Robert Israel's last paragraph, in 6.3.4 it is stated that $C(X)$ is an AW* algebra if and only if $X$ is extremally disconnected (Stonean), and $C(X)$ is a W*-algebra if and only if $X$ is extremally disconnected with a separating family of normal measures (hyperstonean).