I'd say at the most simplified level, "serious" games would be desktop games you install, "casual" games would be in the browser. Partly that's tech-driven though, "serious" games with fancy 3D graphics kind of need to be proper applications whereas a 2D game can easily be done in the browser.
You might do well to identify your 'competitors' i.e. games catering to the same people you want to play your game, and see what they do.
Just a note, but you should be aware of the current trend of browsers moving towards abolishing (traditional) plug-ins based on the NPAPI (Netscape Plugin API) by blocking them. Currently there is a whitelist and I think the user will always be able to enable a NPAPI until support is completely dropped, but it does mean browser plugins that don't support newer plugin api's will be at a disadvantage.
To be honest my knowledge in this area is limited as I rarely do browser content these days, but as far as i'm aware both Chrome and Firefox have taken the above approach and to my knowledge some engines that support browsers have not yet updated to the newer plugin api. Worth doing a bit of research on this to see if it might influence your choice of development engine and wether to go PC or browser or both.
Of course the other problem with supporting browsers through plugins these days is the increasing use of phones and tablets where they are often not supported. That might be something else to factor into your decision. Personally i'd probably pick the engine with the widest platform reach and develop the project with a view to enabling it to run natively on many platforms i.e. PC/Mac standalone, Android/iOS Apps, Browsers etc.
