They are very similar, but wgsl is its own language. Vulkan can run SPIR-V shaders written in glsl, hlsl or slang. Other experimental languages exist (like rust-gpu).

Wgpu is a bit behind on GPU features, for example they've added support for shader subgroups (aka warps or waves) in 2024, where as this feature was available in Vulkan 1.1 released in 2018 or Direct3d 12 shader model 6.0 (same timeframe). Wgpu still does not support buffer device address ("GPU pointers") which I consider quite a game changer.

Many popular tools, e.g. RenderDoc, don't have wgpu support.

If you are not targeting the web platform, wgpu isn't really bringing anything to the table.