Great question!
Both theories are about the movement of Earth's surface but they differ in when the theories were first posed and what Earth phenomena they describe.

Continental drift was coined in the early 1900's by Alfred Wegener, though he did not discover this concept. Wegener saw that the continental landmasses on maps line up like a jig-saw puzzle, as well as geologic features (like mountains and valleys), glacial-caused erosion patterns, and fossil areas line up across the continents for a once connected single landmass.

Continental drift was based on the idea that all the modern landmasses were once connected into a single continent. However this theory was rejected because there was no known mechanism for how the continents could have broken apart and ended up where they exist in the modern time. Then, as World War II was ending, a new radar technology revealed the Atlantic sea floor was spreading away from a mid-ocean ridge in the center of the ocean bottom. Further studies showed the mid-ocean ridge is a chain of volcanoes running down the center of the ocean floor, a magmatic zone where lava comes up from the mantle, creating new ocean floor and pushing the crust apart. Suddenly there was a mechanism for the movement of the continental masses.

Other studies of spreading at rift valleys on continents, subduction zones destroying the outer edges of ocean crust, and the theory of plate tectonics was born. Plate tectonics theory became widely accepted starting in the 1970's, and describes Earth's surface as broken into numerous plates that move around Earth's surface, either slamming into another plate, destroyed under a neighboring plates, or grinding past another plate. The tectonic plates, or slabs of Earth's crust sitting on top of mantle rocks, move as rigid crust on flowing mantle rocks (crust on the mantle is like a graham cracker on a marshmallow, both are solids but the mantle rocks can be deformed plastically like a marshmallow). That movement can cause magma, gas, and ash to shoot up and create a volcano and/or an earthquake when the plates shift and slide.

Earthquakes are caused when two tectonic plates are colliding. They move because of the intense heat from the Earth's core. As the lava cools it shrinks and as it heats up it expands causing the plates to move.

Thus continental drift describes continental crust moving around the ocean basins, while plate tectonics describes how and why Earth's crust is created, destroyed, and moves.