Hi StrawberryBlonde, let me see if I can be of some help. I have been a student at USU for 2 years, but just recently bought a house in Logan. The area is known as "The Island" and it is right below the University's "Old Main Hill". It's a beautiful area, and I have the Logan River running through my backyard. Anyway, you should be aware that Cache Valley is fairly small. You could drive from one end to the other in 45 minutes tops. The Idaho border is about 30 minutes north of Logan. There's a place there called "La Tienda" that is flooded with locals from here before big Powerball lottery drawings. I've heard they sell more lotto tickets than anywhere else in Idaho. Go figure

As per your questions...
Smithfield is at the north end of Logan, on Main Street. It's about a 10 minute drive from USU campus. Smithfield has some nice places, but I'd say most of it are older houses and older residents. That's not necessarily a bad thing, it's a pretty quiet little town (don't go speeding through it, Smithfield cops are notorious for speed traps)
North Logan is anything north of about 1400 North. Most of North Logan is commercial near Main Street. North Logan is between Logan and Smithfield. Anyway, towards the east mountains it has been growing very quickly, and there are some very nice neighborhoods. Hyde Park could be considered the same as North Logan, they're both very similar. If I were to buy a new house, I'd probably look up there.
River Heights is to the south of Logan. I'd consider it a more run-down area of the city. The houses are older, and I think it has a reputation for being a heavy hispanic area. Hope you speak Spanish
I haven't been out to Nibley in a long time, but I think you could consider it similar to Providence. Providence is growing very quickly as well, and there are some really nice neighborhoods there - especially closer to the mountains. There are a couple of houses I have seen up on the mountain sell for $600-800k. They are mansions. Providence is between River Heights and Millville, about a 10-15 minute drive south of USU campus.
Overall, Cache Valley is pretty much the same everywhere. The only difference is the age of the houses, and how well they have been maintained. In Logan, most of the area west of Main Street and anywhere south of 200 North is fairly old.
Where would you be moving from? If visiting here, the best time to check it out is spring, summer or fall. Winter is definitely not the time that Logan is at it's finest. Air quality on bad days in Logan have been rated the worst in the entire country. Worse than LA, NYC, etc. Believe it! If you aren't familiar with "inversion" - you soon will be. Logan may as well be the Arctic from November to March. Last year I recall one morning on campus clocking in at -15F. That's brutal. If you come for a visit, also be aware to be VERY careful driving on US 89/91 south of Logan. The speed limit is 55, but there are a number of roads that merge into it. Almost everytime I drive on this road I see a wreck.
Drop me a line if I can be of any help to you. I'd be happy to do what I can.
Cheers,
RSJ