Resto vs. rimanente
Both resto and rimanente translate to "rest" or "remainder" in English, but they are used in different contexts and grammatical roles. The word resto is a very common, everyday noun used to talk about the rest of a group, an amount of time, or an object, as well as monetary change. On the other hand, rimanente acts primarily as an adjective meaning "remaining", and when used as a noun, it carries a much more formal, technical, or administrative tone.
Resto
The noun resto is the most common and versatile way to say "rest" or "remainder" in everyday Italian. It is used to talk about the remaining part of a group, time, or object, and it is also the specific word for the "change" you receive when paying with cash.
Ho mangiato metà della mela e ho lasciato il resto a mio fratello.
(I ate half of the apple and left the rest to my brother.)
Tieni pure il resto.
(Keep the change.)
Passeremo il resto della giornata in spiaggia.
(We will spend the rest of the day at the beach.)
Il resto del gruppo arriverà domani.
(The rest of the group will arrive tomorrow.)
Cosa faremo per il resto dell'anno?
(What will we do for the rest of the year?)
Rimanente
The word rimanente is primarily an adjective meaning "remaining", but it can also be used as a noun to mean "the remainder". It is derived from the verb rimanere and is generally more formal or technical than resto, often appearing in business, mathematics, or administrative contexts to emphasize what is left over.
Dobbiamo calcolare la somma rimanente sul conto.
(We need to calculate the remaining sum in the account.)
Il rimanente del cibo sarà donato in beneficenza.
(The remainder of the food will be donated to charity.)
Per i giorni rimanenti non abbiamo in programma nulla.
(For the remaining days we have nothing planned.)
Hanno speso la parte rimanente del budget per la pubblicità.
(They spent the remaining part of the budget on advertising.)
Il rimanente della truppa si ritirò verso le montagne.
(The remainder of the troop retreated toward the mountains.)
Summary
In short, choose resto for everyday conversations about the rest of something or when talking about physical change at a store. Use rimanente when you need a formal adjective to describe a remaining quantity, or a technical noun for a remainder in a professional or administrative context.