Note: I originally posted an answer that I was not 100% pleased with, so I have revised it heavily. 1/20/2017
The projection method is not in an exact approximation to the full system in general. There are a few reasons for this, but the most immediate has to do with the boundary conditions. Let's look at the continuous setting first. Consider the unsteady Navier-Stokes problem in a sufficiently regular domain $\Omega$ with appropriate boundary conditions, given by:
\begin{align} \begin{split} \label{UnsNS} \frac{\partial \boldsymbol{u}}{\partial t} -\nu\nabla \cdot (\nabla \boldsymbol{u} + \nabla \boldsymbol{u}^T) + ( \boldsymbol{u} \cdot \nabla) \boldsymbol{u} + \triangledown p = \boldsymbol{f} \quad \textit{in } \Omega \times \lbrack t_0, t_n \rbrack \ \nabla \cdot \boldsymbol{u} = 0 \quad \textit{in } \Omega \times \lbrack t_0, t_n \rbrack \ \mathcal{B}\left( \boldsymbol{u}, p \right) = \boldsymbol{g} \quad \textit{on } \partial \Omega \times \lbrack t_0, t_n \rbrack \ \boldsymbol{u}(\boldsymbol{x},0) = \boldsymbol{u}_0 \quad \textit{at} \quad t=t_0 \end{split} \end{align} Where $\mathcal{B}$ is an appropriate trace operator describing the boundary conditions. For simplicity, let's assume that $\boldsymbol{u}=0$ on $\partial \Omega$. At a step $n$ the standard Chorin-Temam projection method consists of three steps:
1. We introduce an intermediate velocity field $\tilde{\boldsymbol{u}}^n$ and separate the velocity and pressure computations in two parts. 649
We compute $\tilde{\boldsymbol{u}}^n$ as follows:
\begin{align} \frac{1}{\Delta t} (\tilde{\boldsymbol{u}}^n - \boldsymbol{u}^{n-1} ) - \nu \Delta \tilde{\boldsymbol{u}}^n + (\tilde{\boldsymbol{u}}^n \cdot \nabla ) \tilde{\boldsymbol{u}}^n = \boldsymbol{f}^n \ \tilde{\boldsymbol{u}}^n =0 \, \, \text{on} \, \, \partial \Omega \end{align} Note we enforce boundary conditions here.
1. Next, we use the intermediate velocity field to compute the pressure: $$\frac{1}{\Delta t} (\boldsymbol{u}^{n} - \tilde{\boldsymbol{u}}^{n} ) = -\nabla p^n$$ By exploiting the fact that $\nabla \cdot \boldsymbol{u}^n = 0$, we can apply the divergence operator to both sides and formally eliminate $\boldsymbol{u}^n$. We now have a Poisson problem for the Pressure: $$ -\Delta t \Delta p^{n} = - \nabla \cdot \tilde{\boldsymbol{u}}^n $$
2. Apply the Helmholtz Principle to get the final velocity $\boldsymbol{u}^n$: $$\boldsymbol{u}^n = \tilde{\boldsymbol{u}}^n - \Delta t \nabla p^n$$
There are a couple of things to notice here. The first is that we only have a second order operator for the velocity in step (1), and accordingly we can only control the trace operator (and hence the boundary conditions) during this step. The next thing to notice is that during step (2), we have a second-order operator for the pressure. This implicitly introduces a boundary condition on the pressure; in this case we have imposed homogenous <PERSON> boundary conditions for pressure. Such boundary conditions may not be accurate for the true pressure.
Because our final velocity computed in step (3) relies on step (2), our boundary conditions are no longer guaranteed to be satisfied (as we enforced them during the first step). 649