It's not about what you lose, it's about how you lose it.
5th Edition provides a lot of ways to get things that you want from character builds; healing is one of these things that you've already mentioned, and most features from the divine classes can be produced in some way by another character, or at the very least replaced with a passable substitute.
However, you're losing some things nonetheless if you remove divine spellcasters. Assuming you remove any class which would never have suffered spell failure from armor in 3.5 as a definition for "divine" casters, you have eliminated the Cleric, Paladin, Druid, and Ranger entirely.
None of these classes have any one feature that cannot be replaced: the Cleric's healing can be replaced by a bard, even features like Resurrection that were previously unavailable to arcane casters. Druids are probably one of the hardest classes to replace, simply because they get Wild Shape, but even that can be replaced with Polymorph if you've got a Sorcerer, Wizard, or Bard handy.
However, you don't have any spellcaster class left that uses heavy armor without relying on feats, multi-classing, or other ways to gain proficiency. The Eldritch Knight might fill that category if you are willing to accept a partial spellcaster. This leaves you with mostly squishy arcane casters, who can't block for their allies in combat. Despite this, many of the alternatives to things like Wild Shape will come at a later point in character development, at least depending on how you're using them, and may allow an arcane caster to do a little close quarters combat happily.
The fact that you miss out on limited use abilities like Lay on Hands or Wild Shape that provide utilities without requiring the consumption of a spell slot is also worth noting. You also miss out on any spells which only appear on the spell lists that you have not made available, like Reincarnate and a lot of Druid spells relating to nature. While it may be possible to gain some of these spells back with class features that enable access to particular spells, I don't think that there's any way to necessarily undo that without playing around with giving these spells to other casters or making them available as wondrous/magical items in some other manner.
TL;DR: You don't miss much, but you will miss a few things.
Playing a game without divine casters is possible and won't leave you feeling like you have absolutely no ability to do certain things, but it will leave you with a different theme for the game.
Huh, different theme?
Yes. By sacrificing divine casting classes, you've gotten rid of the dedicated healers (barring perhaps the bard, but that's not what bard is best known for), nature-themed characters, and most members of religious orders. You can replace the ranger with, say, the nonmagic ranger from Unearthed Arcana, but even that isn't thematically a replacement for the niche druids and rangers fill in some of D&D's fantasy settings.
Since you're likely running in another setting, you don't have too much to worry about from that, but it is something to consider: you might want to check out content outside the PHB for some nature-themed archetypes if you want to have nature magicians in your setting, for instance.
Removing Cleric: You lose access to a character who has both healing and access to some potentially very useful spells while still wearing heavy armor. Life domain clerics are probably the best healer in the game by a solid margin, and a cleric is able to wade into combat much better than a bard is, allowing them to truly benefit from the increased power of Cure Wounds without making them a target. Undead might be tricky for a party with no clerics or paladins.
Removing Paladin: Paladins are an alternative to the fighter in terms of damage, but instead of the fighter's largely self-focused ability set they have the ability to heal and protect nearby allies with powerful undead and outsider defeating abilities.
Removing Ranger: Not a whole lot of loss in terms of spellcasting (especially versus the loss of the Druid), though you will lose out on some very mobile fighters who have a great utility in nature areas. Beast Master is interesting, but can often be replaced with a familiar gaining class (Pact of Chain warlock or anyone who can cast find familiar).
Removing Druid: Druids have a large unique spell set that will be lost, in addition to the potential utility offered by Wild Shape. They can also be incredibly powerful casters, though in this sense they're not that far off from some wizards with short-rest recovery. Most of druid's bonuses are situational or can be replaced, but not necessarily efficiently.