We investigated, by experiment and theory, the lateral structure of a weak polyelectrolyte brush at various added salt concentrations and chain grafting densities. Model poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) brushes with grafting density gradients were developed for this study by using a novel "Langmuir-Blodgett-deposition-under-compression" (LB\C) method. Fluid AFM images of these brushes indicate that the lateral structure of the brush system is sensitive to both added salt concentration and grafting density. Under low salt conditions, 0-20 mM NaCl, the brush structure shows strong microscopic lateral heterogeneities at high grafting densities; both the width and height of the heterogeneities increase with increasing grafting density but are independent of the salt concentration. As the bulk salt concentration is increased to an intermediate regime, 60-100 mM NaCl, these heterogeneities become smaller in size and number, coexisting with smooth homogeneous regions. At high enough concentrations, 300-500 mM NaCl, the entire surface becomes homogeneous. A simple free energy-based Flory-type argument is presented which explains the essential features of the thermodynamic behavior of the brush system. In the zero-salt limit, relatively few monomers are charged, and the hydrophobicity of the backbone chain drives the collapse/aggregation of the chains. At high salt concentrations, the brush chains become sufficiently charged to overcome the hydrophobic nature of the monomers and stabilize the homogeneous state. However, at intermediate salt concentrations, it is found that the osmotic pressure of the counterions surrounding the charged polymer moieties can be decreased by collapsing the chain structure while simultaneously decreasing the number of charges along the backbone and releasing small ions into the bulk solution. This effect, which we term "osmotic instability", serves to destabilize the homogeneous brush configuration.