Since I'm not a native Italian speaker, I don't exactly know what the difference between hotel and albergo is. Would you mind telling me the difference, please?
The first difference is that "Albergo" is an Italian word and "Hotel" is a French/English imported word. You won't find many Italian words starting with H and the ones you can find are normally imported.
The second difference is a consequence of the first: Albergo is used more by older people. You might know that between the two world wars Italian were "warmly encouraged" to avoid "foreign" words ...
The third is more practical and has already been explained above: often an Albergo is a simpler place. A Hotel often enjoys a 3, 4 or 5 star classification.
Once there may have also been some difference in taxation. Our Tuscan restaurant in Genoa was a "Trattoria" also because my grandparents wanted to save a little money in taxes. I hope that doesn't apply any longer, but in this Country you never know.
In addition to Elberich's answer, I'll also add - but be aware that this might be a bias of mine, possibly due to regional usage (Italian has quite a lot of regional variations) - that Hotel is more often used in proper nouns (possibly due to delusions of grandeur on the part of the management). So you might use both in a sentence:
Staremo in albergo, all'Hotel Edelweiss. We're going to lodge at Hotel Edelweiss.
(You can say "staremo in hotel", but at least in Tuscany you usually say "stare in albergo").
I think the level and luxury of the place is not connected at all with the use of "Hotel" or "Albergo", that are in my opinion synonyms.
"Hotel" is more recent in Italian language, so "Albergo" is used more by older people because the French/English word "Hotel" wasn't used so much in Italy in the past. By the way: this word derives from the Latin word "Hospitium" that in Italian took different branches:
"Ostello", place to spend a night, like "Youth Hostel" = "Ostello della Gioventù".
"Osteria", a small restaurant or place where mainly offer wine.
"Ospizio", house for elder people.
"Ospedale", place to cure ill or injured people.
You can say "Albergo a 5 Stelle di Lusso" and no one will look at you like you said something wrong.