In Tomb of Annihilation, there's a character who offers a spellbook containing some random spells as a reward. My players don't have an arcane magic user, just a druid, so would they be unable to use this item? It's not a big deal if so, I'll just substitute some magic items or scrolls instead.
Sources of Spells
From a mechanical perspective, there is no such thing as "arcane" and "divine" magic anymore. In 5th Edition, they've harmonized a lot of the spellcasting rules each spell has a level, and has identical effects regardless of the class that grants it. Half-casters (Ranger, Paladin) and third-casters (Arcane Trickster, Eldritch Knight) may get a particular spell at a later character level due to slower progression, but the spell works the same way as any other caster.
Using A Spellbook
In order to make use of a spellbook, the character has to have a feature that allows them to copy over spells.
- All Wizards have a Spellbook; an ever expandable collection of knowledge. Given sufficient time, money, and exposure, they can copy any and all spells on the Wizard list into their spellbook.
- Some Warlocks have the Pact of the Tome boon and the Book of Ancient Secrets invocation, allowing them to scribe and cast rituals from any class.
- Any character can take the Ritual Caster feat, and be able to scribe and cast ritual spells from one particular class.
Other Spell Knowledge
Each class has a (sometimes subtly) different way to determine spells known and spells prepared. Druids and Clerics, in particular, already "know" all the spells it is possible for them to know. The other classes methods are beyond the scope of this particular question, but should serve as good reading.
Reiterating small parts of T.J.L.'s answer because I want my answer to be complete, but have important aspects to add.
Divine versus Arcane
Distinctions between the two do not exist. Magic is magic, and even psionic abilities radiate as magic, even if they do not count as spells.
There are only two RAW reasons to use a spellbook:
Wizards prepare all spells from their spellbook.
Characters with Ritual Caster can cast spells tagged with the Ritual descriptor.
Alternate Spell Knowledge
That said, if you desire it to be a worthwhile item, what it can do is introduce players to spells that they haven't seen in the Player's Handbook.
In Adventurer's League, they specifically only allow expanded spells from the "+1" source-book that is chosen during creation, so the concern about issues like spell list bloat are dealt with, and Wizards can learn from outside their "+1" if they can find a scroll with a spell from outside the PHB and their extra source book.
There's no technical RAW method for casters outside of Wizards and ritual casters to gain new spells if they become available. Technically, by RAW, if you include the spell, it is simply an available option to non-Wizards, but logically, they would at least have to be exposed enough to be able to learn them. In the case of Clerics or Druids, conversation and observation with other casters could provide this, but a spell-book of sorts could be a reasonable buy-in for an in game reason to gain access to spells that, for example, come from Xanathar's Guide to Everything that they hadn't seen before.
It makes more sense than saying that the character didn't know that a single spell existed until they knew a ton of new spells did.
If you take this approach, it can make the 'spell-book' a collection of notes by a more academically inclined caster of the right type, and reading through the book would add these spells permanently to the spells available list of the character in question. For clerics and druids, this means they can choose them when they prepare their known spells every day. For classes like warlock or sorcerer this could make them available as a "spells known" choice when they level.