This is a somewhat gray area, because you're talking about semantics, not hard facts.  But here's an overview of the topic:

In practical terms, usenet and newsgroup are usually synonymous, or at least describe parts of the same system.  Technically usenet is the entire public system of servers, content (articles and indexes) and users, while a newsgroup is a single, smaller portion of Usenet.  Here's a set of linguistic analogies:

Dick: "Where'd you get that book?"
Jane: "From the public library system." (Usenet)

Dick: "Where'd you get that book?"
Jane: "From the technology section in the public library on Main St."  (Newsgroup)

Both describe the same information, Newsgroup is just a more granular reference than Usenet.  Saying you got something "frome Usenet" is functionally the same as identifying the specific newsgroup, though the former is less useful for someone looking for the information.

They're the same because in practice (almost) all newsgroups exist in Usenet.  I qualify that because it is technically possible (and therefore likely, somewhere) that a newsgroup server running the NNTP protocol (see below for more information on that) is not a part of Usenet.  For instance, certain government or corporate networks have zero connections to the outside world.  There is no way through data transmission lines (physical or wireless) to move a piece of data from that closed network to the Internet, without copying it to a piece of media and taking it to a diffrent computer at a different location that does have access.  So the machines on the closed network are not part of the Internet, because in the most general sense the Internet is a description of all the sub-networks that connect to exchange information.  Any network without a connection to it is therefore not a part of the system.

If at home I unplugged my broadband router, installed an NNTP server (usenet server software) and set up a newsgroup, and put NNTP clients (usenet/newsreader software) on my other computers, I'd have a newsgroup that wasn't a part of Usenet (nor would my computers at that time be part of the Internet).  I actually did this once upon a time, just for grins so I could understand NNTP better,

In pratical terms that's not very helpful, since only I, my wife, and our dogs could exchange information on it.  But I can see instances (probably on those same "closed" networks) where people might do something like this as a means of posting quickly accessible information in a lean format foe general consumption or discussion.  A research institute might post findings internally so everyone has the same, current data to work from for example.

So for total content exchanged (probably, since we can't measure closed networks, but they also don't matter for anyone not given access to them), 99% of newsgroup data is exchanged through Usenet.  But yes, it is possible to have newsgroups (NNTP servers and clients) running outside public Usenet.

Usenet is one of the oldest computer network systems still being used. As a result of its age, the fundamental concepts of the threaded discussions and community have been recreated many times on the 'net.

Usenet and Newsgroups are synonymous. A newsgroup refers to a specific topic or section of Usenet as a whole.

I really find it interesting how much of the web is currently based on the fundamentals of Usenet. Mahalo is a great example of this. We contribute and ask questions in a threaded method very similar to newsgroups. Many of the most popular sites online are currently based on the same basic community and communication methods that Usenet provides.  From torrent directories to web based adult content collections, much of the inspiration for sites that are currently successful derive directly from Usenet and Newsgroup concepts.