This is a bit difficult to answer, as it depends on many factors.
In general, you can assume that you will get performance close to native apps for the GPU side, as the WebGL graphics API uses your GPU for hardware accelerated rendering - there is just a little overhead for translating WebGL API calls and shaders to your OS graphics API (typically DirectX on Windows or OpenGL on Mac or Linux).
For the CPU side, all your code is translated into asm.js JavaScript. So what kind of performance you can expect depends a lot on the JavaScript engine of the web browser used, and there are some pretty significant differences there currently. At the point of this writing (Jan 2015), Firefox delivers the best performance on Unity code, as Firefox is currently the only browser which makes use of the asm.js spec to use an optimized AOT compilation path of JavaScript code in that case, which delivers performance within a factor of less then 2x slowdown compared to native code for many programming benchmarks - and that factor also matches results we’ve seen from different unity content we deployed to WebGL and ran in Firefox.
There are some other considerations, though. Currently, the JavaScript language does neither support multi-threading, nor SIMD. So, any code which benefits from these features will see bigger slowdowns then other code. You cannot write threading or SIMD code in WebGL in your scripts, but some engine parts are normally multi-threaded or SIMD optimized, and will run less performant on WebGL because of this. An example is the skinning code, which is both multi-threaded and SIMD-optimized. You can use the new timeline profiler in Unity to see how Unity distributes work to different threads on non-WebGL platforms. Longer term, we are hoping that these features will become available on WebGL as well.
For best performance, set the optimization level to “Fastest” in the Build Player dialog, and set “Exception support” to “None” in the player settings for WebGL.
You can use the Unity profiler on WebGL, just like on any other platform. One important distinction is that you cannot attach to running players in WebGL, though, as WebGL uses WebSockets for communication, which will not allow incoming connections on the browser side. Instead, you need to use the “Autoconnect profiler” checkbox in the build settings. Note also that draw calls cannot currently be profiled for WebGL.