My entry point was AD&D and 2E. Main difference there was that Clerics have their churches and cloisters and such with rites specific to their deity and spells granted as part of the divine portfolio, but also with the internal factions and the sort of ecclesiastical politicking one might expect. Just going off what they had in the Complete Priests handbook.
Druids by contrast had a much more uniform organization that was supposed to extend across the realms with a sort of vague hierarchy of circles within greater circles. They were kinda interesting in that the lvl progression included set roles that the player character would then adopt. So for example to be a High Druid meant that you were lvl 12 and one of only 100 druids in all of Faerun under the Arch druids (30) under the Great Druids (9) and then there was only 1 druid in all the realms who could be the Grand Druid. This required a moot or a challenge in order for the player to become that Druid and then deal with whatever current cataclysm was threatening the greater balance worldwide. After lvl 16 the player character becomes a Hierophant Druid. These were the most powerful druids in the realms and they cease to age, but they exit the whole structure and basically go free agent at that point.
In addition to speaking with Animals and communing with plants, Druids also have their own secret language in D&D based on stuff like ogham, to communicate with each other in cypher. I always thought that was a cool idea.
These classes were both part of the priest archetype in the big 4, but the divides back then were mostly down to things like equipment which was tied way more to class than it is now. You know stuff like Clerics using blunt weapons and heavy armor, whereas druids got the bladed weapons like scimitars and daggers but had to keep it in leathers. Then their spell lists also reinforced the division by type, so Druids get different sorts of immobilizing spells and summons spells. All that elemental flavor and the animal pals.
Clerics I always got a vibe like that was more flexible for what might constitute their churches and such, but Nature Cleric might have seemed a bit curious since there's so much overlap. I kind figure that those would be the Ranger/Clerics or the Fighter/Druids of previous editions, which were the two multi/dual class combos that sorta filled out that niche. Sorta like Bards would sub in for Mages on the arcane front, or Rangers might sub in for Rogues for the scouting/utility stuff.