Questions: How to subscribe to a new tag? What is the meaning of "Watch tag"?
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 directly on the site.
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 filter. (And also set up how often you want to receive an email with updates.)
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'
Some basic information about tag-watching can be found here: What do “favorite tags” and “ignored tags” do?
Subscribing to filters
If you create a filter, you can choose whether you want to receive notification and set up some basic settings - namely the frequency of the updates:
After you do this, you will receive an email asking you to confirm the subscription. (Basically to check whether you did not make a typo in the email address and whether the email address does indeed belong to you.)