Flash allows you to draw things and then make actions happen to those things using action script. Iirc (it's been a while) you can do this within a very user-friendly GUI and relatively little code, making things nice and easier for a designer. With HTML 5, you design in sentences, which doesn't suit everyone. Yes you can embed videos and animate drawings jscript. However you need CSS also. So that's 3 languages you need whereas with Flash, you just need to be handy with frames and the pen tool and learn action script. (Though I suppose there aren't many action script people that don't know CSS also).
I'd say that most companies who develop games using Flash don't use Flash Pro, they use Flash Builder. Flash Builder, which is based on Eclipse, is an IDE specifically suited to building ActionScript applications. There's no drag-and-drop GUI, you get a code editor. The only relationship between Flash Builder and Flash Pro is that the end result is played back using Flash Player.
If you're developing a game with Flash, Flash Pro becomes quite cumbersome to use once you get past one or two scenes and a few buttons.
With HTML5, there are a few drag-and-drop GUI's you can use. The most prominent of which is Adobe Edge, Adobe's own HTML5 animation tool. It does a lot of the stuff that Flash Pro does, but not all. Although it's still in the alpha/tech preview stage, people are using it for actual production work, and there's even a game on the iOS App Store that was developed with it.
Personally though, if you want to build games with HTML5, you certainly want a code editor, of which there are many HTML/JavaScript/CSS ones out there.
It's also worth noting that you don't need CSS to do the animations. A lot of people use plain old JavaScript for it (Myself included), by either using or implementing their own tweening library. Although it's a good idea to mix them and/or switch to them when appropriate, as certain devices provide hardware acceleration for CSS3 transforms.
To answer your question ("Is Flash ONLY good for browser games now"), I'd say no, but that's pretty much the only thing it's being used for aside from video players, ala YouTube. HTML5 is still a ways behind in terms of capabilities, but you can still do some pretty impressive stuff with it, to the point where I think people should be asking "Can I do this using HTML5 instead of Flash" instead of sticking with what they know.
EDIT: I will say that one of the advantages of Flash Pro is that your artists and designers can animations quickly and easily, and export them as a swf. This makes it really easy for the coders to use the ready-made animation in ActionScript by importing the swf to the stage in code in Flash Builder.