This is how I understand it:
A secret admirer...I think they fall into two categories: those who may follow you a bit, and those who won't follow you at all. In either case, the main point is that they fancy you but they don't know how to approach you properly, either through shyness or ignorance. This is where you get the sort of engineered "Oh, I just happened to be passing in this hallway" kind of thing as an excuse to talk to you. Alternatively, they might send you an anonymous valentine card or something like that, but they won't follow you in any way and will simply be awkward around you.
A stalker has malevolent intentions. They follow you around everywhere: online, on the street, by text, EVERYWHERE. It's very intense, invasive, and predatory, and these people are clearly dangerous. Their behaviour can't be explained as a shy person who doesn't know better. It's too calculated and too determined.
In short: if a secret admirer follows you around it will be annoying but not much more than that. If a stalker follows you around, it will be intense and invasive and they will be very hard to avoid. It's the quality and the degree of the following more than anything that identifies the stalker.
As for your question about sending an SMS message while remaining anonymous...it could be interpreted that way. At the very least it is bad manners and will leave them feeling uncomfortable. If this wasn't a hypothetical question and you're considering it, then it may be better if you explain to us what you're trying to do and why and then we can suggest a better way to do it that will be less risky.
Merry Christmas!