I think there are potentially a couple of different issues here. First, Tableau Public and Server can often display slightly differently than desktop. I've found this to be a particular issue with text. So, you end up just having to tune it and tweak it until it looks just right on the Server (that's where you'll share it anyway, right?). Second, you're using some non-standard fonts. Tableau doesn't do a great job with fonts unless they are supported fonts. So, often the browser will convert these to a different font. This will particularly be an issue for any users who don't have that font on their machine. My recommendation, therefore, would be to switch to one of the Tableau family fonts and then see if that resolves the issue.
Here are some details on specific web-safe fonts: Tableau Online/Public Browser-Safe Fonts