These two styles get conflated constantly because both favor clean lines and natural materials, but the philosophy behind each is different.
Mid-century modern (roughly 1945-1969) is defined by horizontal lines, exposed structure, large glass walls connecting indoor and outdoor space, and a deliberate lack of ornamentation.
Modern farmhouse borrows agricultural vernacular and pairs it with contemporary open-plan interiors. It is fundamentally a hybrid style: traditional silhouette, modern function.
The easiest tell when looking at a real building: mid-century modern rarely uses a steep gable roof, while modern farmhouse almost always does. If you are renovating and want a coherent result, pick one structural silhouette and let your material choices reinforce it rather than fight it.