What's the difference between a photographic lamp that is always on and a flash?

The biggest difference is color temperature and illumination.

First, the flash illumination is much higher, and the corresponding shutter speed is much faster. The lamp illumination of constant light source is low, and the shutter speed required is much slower.

Second, take a portrait photo or flash. If you just shoot still life, it is enough to use tripod+shed, even with ordinary tricolor light. After all, the subject is absolutely stationary, so the shutter speed doesn't matter.