The M4 Mac mini will be much, much faster, provided that it has enough RAM for the workload you are running.
I'm making assumptions here, but if your old MacBook was running Catalina, and that was "the end of the line", we would be talking about a machine with a model year of maybe 2012 or 2013. Let us use a MacBook Pro (Retina, 15-inch, Late 2013) with a 2.6 GHz quad-core Core i7, and a discrete NVIDIA GeForce GT 750M GPU, for our baseline.
- Single-core CPU benchmark score: 1,063 vs. 3,748. (M4 Mac mini's score is about 3.5x higher).
- Multi-core CPU benchmark score: 3,598 vs. 14,467. (M4 Mac mini's score is about 4x higher).
- Metal (GPU) score: 1684 vs. 56,887. (M4 Mac mini's score is about 33.8x higher).
- FP32 (GPU) score: 742.7 GFLOPS vs. 4,260 GFLOPS. (M4 Mac mini's score is about 5.7x higher).
If we look at ports, that MBP would have USB-A (USB 3.0) ports, a HDMI port, and Thunderbolt 2 ports that are now useful mainly as Mini DisplayPorts. A M4 Mac mini would have USB-C ports that supported USB 3.1 Gen 2 (up to 10 Gbps), USB4 20 and 40 Gbps, and Thunderbolt 4, allowing for use of very fast external SSDs (if desired).
The SSD built into the M4 Mac mini is also likely to be faster than the one built into a Retina MacBook Pro. (If by any chance, your old MBP is one that starts up off a mechanical hard drive, there will be no comparison at all.)
I would suggest reconsidering going with only a 256 GB SSD. Even if you plan to put most bulk data on an external drive, you probably want at least a 512 GB SSD. With regard to RAM, you should order as much as you need, or are likely to need, as you cannot upgrade RAM after purchase. The plain M4 Mac mini is available with 16, 24, or 32 GB of RAM – and if you are thinking of running, say, the Adobe suite, you might want more than 16 GB.