Programmer just writes code.

Software engineer designs the system that is to be coded.

It’s like the difference between an architect and a construction worker.

Programming is just one part of the whole process of creating software. There's more than just writing code. There's also planning, design, architecture, testing, deployment, maintenance, etc. All that is engineering. Unsurprisingly, people with software engineering training tend to have a more complete idea as to what goes into it all.

Calling yourself "engineer" impresses more people and gets you higher wages.

Let’s say I’m a carpenter, and occasionally I open up a cad program and draw up a wrench or lathe chuck and have it cut out of sheet steel. Yes, I did some engineering but I’m not really an engineer.

Likewise let’s say I’m an artist. Maybe I write code now and then to program my automatic paint mixer or whatever.

Code to build something: engineer

Code to use something: programmer

To me, if you wrote/maintain/design a scaled system with uptime and latency requirements, that's a software engineer.

If you're laying out buttons and implementing business logic, you're a programmer. (This was me).