You are correct - somewhat. Sprites, by default are rendered directly against the camera, however you can easily change this if you are using the Sprite Renderer in a 3D scene.
Sprites are physical objects in your scene, whereas Texture2D is exactly what it says it is. A texture. A texture must be attached to a material, and the material to a game object(e.g a plane).
Back in Unity 3.x days you didn't have sprite support right out of the box so you had to roll your own Sprite Manager/Class(or you had the option of buying an asset off the asset store that attempted to remedy this annoyance). Unity was really never meant to support 2D games(nevertheless developers found ways to make it happen), until 4.x, when the Unity Developers finally provided built-in support for 2D games(sprites, sprite sheets, 2D physics).
If you're doing a 2D game, always try to use the Sprite class. It's better than rolling your own, and offers more than enough.