This is only partial answer, since I was unable to replicate the behavior you describe. (I do not see the "manage subscriptions" button. Possibly the reason that I do not get this button might be that I have set up a new filter subscription just now to test what you are describing your question? Or perhaps this works only for tags which have been subscribed before favorite tags were changed to tag watching?)
As far as I can tell, the difference between subscriptions and tag-watching is that the former provides you with possibility to get updates about content via email and the latter changes the way you interact with the questions having these tags.
How to subscribe to a new tag?
You can add a new subscription in the page with filters. You can create new filter (or use some of the already existing ones) and choose whether or not you want to receive emails with new questions from this tag. (And also set up how often you want to receive an email with updates.)
You can see all your subscriptions in the subscriptions tab of your network profile. (It is also possible to unsubscribe and change settings from this page.)
What is the meaning of "Watch tag"?
Tag watching is simply a new name for the feature which used to be called favorite tags: 'Favorite Tags' is now 'Tag Watching'
As pointed out in this answer, before this change there used to be choice to subscribe/unsubscribe on the tag pop-up before this change. The same thing was mentioned in this recent question: How to subscribe to tags?
Some basic information about tag-watching can be found here: What do “favorite tags” and “ignored tags” do?