After reading the materials and digging into a little bit of background, my opinion of this situation is as follows ...
I see a person who has gained a bit of celebrity status and isn't happy about all of the things that come with it. Rather than shrugging off the negative and moving on, they've let those things get to them and affect who they are. I also see a hypocrite who is complaining about being sexualized when their fame is due, in part, to sexualizing themselves.
It's more of a bait-and-switch than hypocrisy.
Frankly, I didn't really pay much attention to Rebecca Watson and Skepchick before this bruhaha. At first, I was pretty strongly on her side for the initial reactions, which seem to me still to be pretty moderate. However, the more I've looked for the things she said and did, the less impressed I am with her. Other than using sexualization to promote herself, she does not appear to be very talented.
As for Dawkins, I gave up on him a long time ago. His comment was certainly rude and inappropriately focused on Watson, and I am guessing that there was some alcohol involved. It felt to me that it had the miasma of ethanol. I think I could have made the point much better, even if I were drunk.
Because he does have a point. The point is about privilege. That is the title of Watson's response, in which she calls Dawkins a rich white male (what color does she think she is, anyway? Purple?)
As Watson frames it in terms of privilege (Dawkin's versus hers), it is worthwhile to point out how much privilege that she has. This is not only how she gets to be on a panel with someone whose career started before she was born, but how privileged Western women are to the status of women in many middle-East, Islamic countries (or even in the West, where there have been a few "honor killings").
This is particularly important as Watson invokes feminism, and for the past three decades or so, there have been a lot of feminists with a rather idealistic notion of what women face in places like Afghanistan and Saudi Arabia. The only major feminist (if you want to call her major) who has seriously bucked the trend has been Phyllis Chesler. When young, she married a (rich) Muslim man and lived in Afghanistan. So she has some personal experience with how restrictive such a life can be.