Why is looking the same what you want? Future fonts will probably suit future changes in display technology (e.g. the shift back to serifs as screen density has increased) and/or future shifts in written language (compare blackletter or Fraktur style to today's scripts). If you want your website to be readable in the future, letting future platforms display it with their default font seems far more dependable than trying to control how it looks pixel-by-pixel.
Serif fonts have looked more or less the same since 1465. [1] Blackletter fonts didn’t last very long, because they were not very legible, and were quickly replaced.
Sans Serif (Grotesque) for formal printed material is newer, probably around 200 years old, but is also widely established. Regardless, on my personal website, I use serif lettering for most text, because while it’s slightly less readable at 75dpi, retina displays are common enough (e.g. pretty much any phone in use today has retina level resolution) that serif text makes more sense. [2] Also, the only open source font the great Matthew Carter (Verdana, Georgia) has made is a serif: Bitstream Charter and its derivative Charis SIL.
The English language is much more likely to change to the point today’s English can only be read by linguists and researchers before typefaces undergo any significant change.