"I have very good connections."
Translation:Ich habe sehr gute Verbindungen.
Its like the difference between "really good" and "very good" in English. It isnt a very big deal which one you use as they are nearly identical. If you are in a conversation however, you should switch which one you are using every once in a while to avoid sounding too repetitive.
They are similar but don't mean exactly the same. It's a bit like the difference between very and pretty, with sehr being the word that is... Stronger (in lack of a better word to explain it). For example, if you ask somebody: Wie geht es dir? And they respond with: Ganz gut. They probably mean that they are good but it could still be better. If they say Sehr gut it usually means that everything is as good as it could be In addition to that ganz is something you'll hear more often in the spoken language than in written (official) texts.
Verbindung is a lot more versatile than Anbindung. Personally, I would only use Anbindung to speak about technical and maybe public transport, while Verbindung is a word that I use practically every day (it works for technical stuff and public transport too, but you can also use it to say that you know a lot of important people (which was my first interpretation of this sentence) and a lot of other things).