There is a lot of overlap but some differences in emphasis. Control Engineering is also older than DSP. If you have a traditional EE education, you don’t really make much of a distinction.
State variables are the more typical perspective in Controls. The first edition of Oppenheim and Schafer 1975, had a chapter on state variables, but they dropped it over the years. You need to understand state variables to do Kalman Filtering which is an area of overlap. Linear Estimation and Linear Controls are duals of each-other.
I would also say that hybrid continuous/discrete time systems are more common in Controls but there are many examples for DSP as well.
DSP is almost always done on uniform sampling. State Variables can work with nonuniform sampling as well.
I’ve never heard of anti causal Control System but forward backward filtering in time is common in DSP. Controls are inherently causal. The one sided Laplace transform is more common in controls.
Stability in feed back loops is important in both areas. An advanced control systems class will cover topics like Lyaponov stability. You typically don’t see that covered in DSP but there are DSP papers that use that technique.
Control Theory shows up in mechanical engineering. DSP shows up in finance. There is a-lot of both in robotics which also uses computer vision.
In RADAR, waveforms and filtering are more DSP at the front end, but the tracking systems at the back end are more Controls like.
If I had to use a single word to describe each.
Controls: feed back
Signal Processing: sensing
or maybe using a phrase