When do you use a table in HTML vs more "modern" approaches?

table elements are still the easiest way to make a table, and in a lot of contexts the best. If the demands of the interface are not too complicated, I'd personally default to tables.

I'm not strongly-opinionated though, so want to see if others disagree.

If you want a search engine like Google to understand your tabular data, use a table.

Even fancy layouts like here on our stock instrument table are totally possible with a classic table:

Mail templates are 100% table oriented.

I use table element to create a table. If it is not a table, I don't use it. If what I am going to display has data on fixed location, header, and can be easily expressed as a table. I use table tag for it. By the way, table is not suitable for small devices. I would change to card based display instead.

If I want to present data with more than one-on-one relations (which could be a definition list), I will use a table. For one-on-one relations, it depends on the use case, so I might or might not choose a table.

Don't forget that a table is HTML and modern layouts like grid are CSS. It's not either/or. You can style an HTML table with grid if you want to.

Use the is it a duck theorem, if it looks like a table it should be a table

Which are the modern approaches?

Accessibility and now let me know what your "modern" approaches" are.