They are not the same, but you mostly have to have played TT and be a bit of a lore nerd to understand the fundamental differences. All except the faithless in the forgotten realms worship a god; worship alone doesn't mean much.
Clerics are vessels of divine power who gain their power directly from a single god. They work toward their god's agenda and take on their alignment of their god. Radiant damage is not light, it is holy power and more akin to fire; the source books often refer to it as "holy fire." But clerics can also do necrotic damage. The spell "Heal" is not by nature good, it is neutral: it can be used to heal good or evil people. Regarding Shadowheart's alignment, she is a cleric of an evil god. Part of her character conflict is that her alignment doesn't match Shar's, so much so that Shar specifically punishes her for it.
Druids gain their power through connection with nature itself and serve the concept balance. Their connection with nature is so deep that the most powerful of druids have dramatically slowed aging. Death and decay are as natural as new life. Those are neither good nor evil from a druid perspective. If something is unnatural, like mindflayers or undead, they are against it. Druids are traditionally true neutral, although that has changed in the underlying rules, and I think it's fair to argue Halsin is neutral good. Aside from Halsin's comments and the existence of spore druids, BG3 doesn't really delve into this.
Mechanically speaking, nature clerics try to be both and that's why they are C tier at best.
Ultimately, as Gray Ghost says, it's about power fantasy, the same as any other class. All the classes in DnD share commonalities with others. If we are looking at similarities, you might as well argue that Cleric should be a subclass of Paladin.
