In 5th Edition, the primary source of information about both planes is the DMG. Since you're asking about 5th specifically, we'll avoid talking about earlier editions.
The Ethereal Plane is what connects the Prime Material Plane, where all elements are found, to the Elemental Planes, which serve as the sources of those elements. While in the Ethereal, you can perceive the Material Plane if you're close enough to it, a region known as the Border Ethereal. Think of the various planes of matter and energy as a sphere, with the Prime Material as the globes surface and the different Elemental Planes bobbing about inside it. A mirror image of the Prime Material serves as the Feywild, and the shadow the Prime Material Plane makes is the Plane of Shadow.
The Astral Plane, on the other hand, is a timeless void. Everything in existence (except, due to the weirdness of cosmology, the Inner Planes) is accessible from the Astral Plane. Unlike the Ethereal Plane, travel in the Astral Plane is tied directly to thought, and it is believed that intelligent creatures travel to portions of the Astral Plane as they sleep. Near the outer areas of the Astral Plane, itself an eyeboggling concept due to its infinite size, are the realms of the abstract where the gods live and souls travel upon death, the Outer Planes.
Your key takeaways:
The Ethereal Plane can look upon the closest plane. It is otherwise a foggy realm that connects the Inner Planes and the Material Plane. It's not true to think of the Inner Planes as inside the Material Plane and the Ethereal as the goo holding it all together, but it helps.
The Astral Plane is timeless, almost featureless, and is home to everything non-physical or beyond mortality. The Astral Plane is what the Ethereal Plane sits in, surrounded at a great distance by the Outer Planes.