If something is conceptual, it has to do with underlying ideas or thoughts, while the meaning of something contextual depends on its specific situation or circumstances.
Conceptual means characterized by thoughts or concepts, from the Latin, conceptus, "a gathering." When you're at a project’s conceptual stage, you’re still gathering ideas about how to go about it. And in conceptual art, the artist’s ideas are more important than any physical artwork they create. Conceptual is all about abstract thinking:
But there are many more methods of planetary defense; it’s just that, for now, they are purely conceptual. (Salon)
Mathematical theories were not explanations, they were conceptual systems for making predictions. (The Invention of Science)
The Latin root of contextual means "to weave together," the way you weave in contextual details like when, where, or why something is happening in order to clearly explain it. Without the contextual information that it's Talk Like a Pirate Day, people will wonder why you're greeting them with a hearty, "Ahoy, matey!" For more context:
It provides a contextual history — enough details on economic crises and the political fortunes of various prime ministers to test even the most loyal fans. (Washington Post)
"Without any contextual information and expertise, the general public will not be able to really understand what is in those files," he said. (New York Times)
To confuse matters, there’s a third, less common word that’s similar to conceptual, describing the initial spark of an idea: conceptional. It’s also a scientific term referring to biological conception, when an embryo is conceived at the beginning of pregnancy.
Embryoscopy can be performed as early as 3 conceptional weeks’ gestation and continued throughout the first and into the second trimester. (The First Twelve Weeks of Gestation)
Layman said the housing projects are still in their conceptional stage but are expected to break ground next summer. (Alaska’s News Source)
When choosing between conceptual, contextual, and conceptional, ask yourself if you're describing something that depends on the context for clarity. If so, it's contextual. If you're getting deep, talking about broad ideas and concepts, you should use conceptual. And when you’re focusing specifically on the origin or conception of an idea or an embryo, conceptional is the word you want.