What is the difference between "tool" and "instrument"?

I think entertainment tool is better because it leaves no ambiguity to what you mean. If you say entertainment instrument, there is a possibility that it can refer to a musical instrument.

Your context, however, rules out that possibility. In which case only usage can give you a guideline:

For your specific question you will find answers here.

In general situations, they are the same. However, each of this word has a specific context linking to it. Instrument is used specifically for musical instruments, and tool is for mechanical things such as a tool box.

Besides, in formal/business context, you'll see instrument more (e.g. "financial instruments").

First, the sentence written like that doesn't convey the meaning it's intended to convey. Sounds like the internet needs entertainment, not the other way round. A better rephrasing might be The internet is [the word you are looking for] that entertains.

Now coming to the OP's question, I think neither of two words sounds appropriate in the given context. Tool and instrument, and for that matter device, are used for objects that aid in accomplishing a task— a knife is a tool for cutting, a vernier calliper is an instrument for measuring lengths accurately, and so forth. They are used almost interchangeably though a subtle difference does exist.

In contexts where the effect is sweeping or broad like the one at hand, it would make more sense to go with a different word. I suggest means or agency.

NB : You might want to consult a good dictionary to understand the subtle differences in the meanings of all these words.