As for semantic correctness - it's in the HTML spec, and every major modern browser handles it correctly. Sometimes I wonder if Google's the only folks who actually read the W3C specs, there's been so much cargo-cult advice passed down between web developers.
The "always close your tags" advice came from the early 2000s, when people were pushing XHTML as a way to make your HTML pages XML compliant (the big buzzword back then). It gives essentially no benefit to users, no benefit to developers, costs you bandwidth, makes your pages slower, and clutters up your markup.