I'm playing in a D&D 5e campaign with a bard and a wizard.

Our most recent session ended while we were in the middle of exploring a dungeon. The DM informed us that we leveled up from 4 to 5.

The wizard and bard were excited to get level 3 spell slots for the first time. They began discussing the level 3 spells they were looking forward to casting at the beginning of the next session. I asked, "Don't you guys need to have a long rest before you can cast new spells, though?"

The bard argued that he doesn't need a long rest, because he learns spells immediately upon leveling up and doesn't need to prepare them.

The wizard argued his "number of spells available" count increased immediately upon leveling up, and it wouldn't make much sense if his actual number of prepared spells stayed the same.

Ultimately, the DM decided that it would be more fun if everyone got access to their new toys at the same time. He permitted the wizard to add a level 3 spell to his prepared spell list.

Who is right? Do bards (and other casters that do not prepare spells) get access to new spells more immediately than wizards (and other casters that prepare spells)?

I looked at Can I use my newly gained limited-use ability right after leveling up?, but "Can a mage use his newly gained 2nd-level spell slot?" seems like a separate matter from "Can a mage use a newly gained spell?".

I agree with the DM's call, but spells can only be prepared after a long rest, rules as written. The Wizard would gain his new spell slots, unused and ready to up-cast a spell he already had prepared, but not have a chance to actually prepare the spell without some downtime. This also applies to other prepared classes, so Clerics and Druids are in the same boat.

You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.

The Bard, and his spontaneous friends the Sorcerer and Warlock, do immediately benefit from their new spells and can cast them immediately. They have no need to separately prepare spells and can cast any spell they know as long as they have the slots to do so. (Oddly enough, I can find no direct rule support for this, only that the Bard spellcasting section makes no mention of needing prepare spells like the above quote does for the Wizard.)

https://www.dndbeyond.com/classes/wizard#ClassFeatures

https://www.dndbeyond.com/classes/bard#ClassFeatures

All that said, the DM probably made the right call, if everyone is leveling up between sessions it's not going to be much fun to be the one person who can't use his shiniest new toy. (Personally, I'd probably only have them level up during a long rest anyway, or if I had to grant it during a shorter break for some reason, count that as a long rest, if the adventure could possibly cope with the PCs healing and refreshing powers at that junction.)

Memorized Spellcasters get their new spells immediately; Prepared Spellcasters must wait for a Long Rest

begin{array}{|l|l|}
hline
text{Memorized} & text{Prepared} \ hline
text{Arcane Trickster (Rogue)} & textit{Artificer (Revised, UA)} \
textit{Artificer (Original, UA)} & text{Cleric} \
text{Bard} & text{Druid} \
text{Eldritch Knight (Fighter)} & text{Paladin} \
text{Ranger} & text{Wizard}\
text{Sorcerer} & \
text{Warlock} & \ hline
end{array}

Each of the Memorized spellcasters have the following passage in their Spellcasting feature paragraph, pending a few discrepancies depending on how many spells the class gets in total.

You know four 1st-level spells of your choice from the [class] spell list.

You learn an additional [class] spell of your choice at each level except at levels [...]. Each of these spells must be of a level for which you have spell slots. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.

Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list, which also must be of a level for which you have spell slots.

—The Common Spellcasting paragraph for Bards, Sorcerers, Warlocks, etc., Player's Handbook, pg. 45-120

So as written, the character would gain these spells immediately at the time they level up.

Prepared spellcasters, meanwhile, are forced to wait for a level up:

You prepare the list of [class] spells that are available for you to cast, choosing from the [class] spell list. When you do so, choose a number of [class] spells equal to your [Spellcasting Ability] modifier + your [class] level (minimum of one spell). The spells must be of a level for which you have spell slots.

For example, if you are a 3rd-level [class], you have four 1st-level and two 2nd-level spell slots. With a [Spellcasting Ability] of 16, your list of prepared spells can include six spells of 1st or 2nd level, in any combination. If you prepare the 1st-level spell cure wounds, you can cast it using a 1st-level or 2nd-level slot. Casting the spell doesn't remove it from your list of prepared spells.

You can change your list of prepared spells when you finish a long rest. Preparing a new list of [class] spells requires time spent [...]: at least 1 minute per spell level for each spell on your list.

—The Common Spellcasting paragraph for Clerics, Druids, Wizards, etc., Player's Handbook, pg. 45-120

So a prepared spellcaster would not gain their new spells until they take a long rest.

Well... Almost. There are exceptions for Prepared Spellcasters

Many prepared Spellcasters gain additional spells automatically as they level up. For example, a Land Druid has a list of spells they gain automatically at levels 3, 5, 7, and 9. These spells are gained immediately upon level up, no need to wait for a long rest.

Once you gain access to a circle spell, you always have it prepared, and it doesn't count against the number of spells you can prepare each day. If you gain access to a spell that doesn't appear on the druid spell list, the spell is nonetheless a druid spell for you.

—Circle of the Land, Player's Handbook, pg. 45-120

Additionally, while Wizards would not get to add any new spells to their Prepared Spells list, they would get to immediately add any new spells to their Spellbook. Because Wizards are ritual casters, and are not required to have their Ritual spells prepared to cast them as a ritual, a Wizard would be able to cast any new Ritual spells immediately, without needing a Long Rest.

You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don't need to have the spell prepared.

—Wizard, Player's Handbook, pg. 45-120

All spellcasters gain their new Spell Slots immediately, if applicable

The rules in the Player's Handbook for each spellcasting class indicates that a long rest is required to recover expended spell slots, but there's nothing written that says they do not immediately gain access to new spell slots they attain.

So if a Moon Druid (does not gain extra spells at 3,5,7,9) goes from level 4 to level 5, they would not have access to any new spells; they would, however, have two new 3rd level spell slots to use, even if all their other slots were already exhausted, without needing to take a long rest.

The [class] table shows how many spell slots you have to cast your druid spells of 1st level and higher. To cast one of these [class] spells, you must expend a slot of the spell's level or higher. You regain all expended spell slots when you finish a long rest.

—The Common Spellcasting paragraph for all spellcasters, Player's Handbook, pg. 45-120

Warlocks have some special rules, but are otherwise identical to Memorized spellcasters

At least within the context of this post. I just want to call them out specifically because they have the ability to recover spell slots on a Short Rest, instead of a Long Rest. So because of this, like other Memorized Spellcasters, they gain their new spell(s), if applicable, immediately.

However, their new spell slots is a different issue. Their spell slots "upgrade" to the higher levels instead of gaining new spell slots, so unless a Warlock actually gains a new spell slot (like at levels 2, 11, or 17) their upgraded spell slots would not be refreshed, and they'd still have to take a short rest to gain those slots back.

Technically the Wizard wouldn't get their new spell prepared as:

You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.

And upon leveling up spells are only added to your spellbook:

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook.

Whereas Bards simply learn their new spells and have them prepared:

You know four 1st-level spells of your choice from the bard spell list. The Spells Known column of the Bard table shows when you learn more bard spells of your choice.

However it is not unreasonable to let the Wizard prepare one of their new spells upon leveling up as it lets everyone play with their new powers at the same time, and it means the time the players gained a level (chosen by the DM) doesn't create a disparity between player characters. This problem is obviously eliminated when granting the level increase along with a Long rest.