An invoice is a request for payment that contains information necessary to remit payment to the issuer. Normally that would include the issuer's name, address, terms of payment and if necessary an account number. The invoice also is given a number - a unique key identifier with respect to the issuer.
Later, this number can be referred to in future correspondence. A bill may or may not include any of the above, but an invoice usually will.
Rather, the distinction between these two words is more a matter of custom and the business in question being transacted. For example, a lawyer bills a client, for billable hours. In the abstract, "you should pay your bills on time." Work that is commissioned will usually generate an invoice.
Bill in the sense we're discussing is Anglo Saxon and dates from the 1400s, and invoice is French (envoyer, "dispatch") from the 1500s. You might make the case that invoices are attached to things that are delivered, but in practice it seems that various trades over time have selected one term or the other by tradition and custom.
Notice that in your own examples, you vouched for a person, but guaranteed an action. This is exactly how I'd use these words, and it does seem to reinforce your intuition: "vouch" is about a person's character; "guarantee" is about actions or things. We might also say, however, that "guarantee" emphasizes the future, while "vouch" emphasizes the past. To explain my position, I'll call on the Oxford English Dictionary.
Guarantee
The OED lists the following meanings for the verb "guarantee":
- To be a guarantee, warrant, or surety for; spec. to undertake with respect to (a contract, the performance of a legal act, etc.) that it shall be duly carried out; to make oneself responsible for the genuineness of (an article); hence, to assure the existence or persistence of; to set on a secure basis. (Example: "Written languages guarantee a systematic pronunciation.")
- To engage to do something; to warrant or ensure that something will happen or has happened. (Example: "I'll guarantee that he'll never return to Bengurry.")
- To secure the possession of (something) to a person, etc. (Example: "Christ guarantees to the faith of His brethren...a true quittance and defence from sin.")
- To secure (a person or thing) against or from (risk, injury, etc.); to secure in (the possession of anything). (Example: "Angus was strongly disposed to make the attempt, if he could be guaranteed from loss.")
All of these definitions and examples state that something will happen in the future, or promise to do something or cause something to happen. In the first example, "Written languages guarantee a systematic pronunciation," the claim as I understand it is not that written language can't exist unless a systematic pronunciation is already in place; the claim is that written language tends to create a systematic pronunciation because we come to associate certain symbols with certain sounds. (Obviously, this claim would not apply to all writing systems.)
"Guarantee" is also often applied to products ("We guarantee this vacuum cleaner for life"); again, this is a statement about the future: the vacuum cleaner will not break down, and if it does, the manufacturer will replace it.
Vouch
"Vouch" has quite a few meanings, many of them archaic and/or specific to the legal realm. "Vouch for", however, has only these three:
- To speak or bear witness in behalf of (a person); to be surety or sponsor for. (Example: "I dispute not the lad's qualities, for which your reverence vouches.")
- To supply evidence or assurance of (some fact). (Example: "The certainty of the law...is vouched for...by the results of experiment.")
- To give personal assurance of the truth or accuracy of (a statement or fact). Also with accuracy, truth, etc., as object. (Example: "I love the country better than ever, I can vouch for that.")
I take it that the first meaning is the one you had in mind. My belief that "vouch" emphasizes the past, rather than the present, rests on the fact that even though you vouch for a person with an eye to the future--securing a job for her, for example--you're vouching for certain qualities, like character, experience, expertise, or personality, which are already possessed by that person. You're not guaranteeing that you'll take over the job if she fails; you're just saying that she has the expertise and background necessary to succeed at it.
Best Answer
The only useful distinction I can make is that connection refers to the linkage between two things (nodes) while interconnection refers to multiple connections among multiple nodes. This is not borne out by examining dictionary definitions, however, and if dictionaries (i.e. current usage) have it right the two are pretty much interchangeable.
If you look at the word roots, interconnection seems pleonastic: con (together) + nectere (bind) already means bind together in connect. Adding inter (between, among) to "bind together" only adds "between" or "among" to the meaning. One wonders how things can be bound together without being "bound together between" — but that is language for you.
Etymonline tells us that interconnect is by far the more recent coinage, first attested in 1865, some three centuries after connect. So I really don't have any good advice other than to mention how I think of the difference, as I did in the first paragraph of this answer.