Most of what makes the HP special for me has already been posted, though I can add a couple of things, plus a pic of mine with Spegal grips:
While I appreciate the modern striker-fired poly handguns for their lighter weight and quick-into-action function, I like the simplicity of the Browning HP; detail stripping is pretty simple. Parts are available and likely to remain so.
Finally, "feel" is a personal thing - some people put great importance on this and I'm one of them. For my hand, it's perfect, and that gives me great confidence. In one aspect it's better than any other handgun I've ever handled - the grip length. My middle finger nestles up against the trigger guard while my little finger indexes against the magazine floorplate. This adds to my confidence as I can be sure I'm gripping it the same way, ever time. If it were another quarter inch longer or shorter I wouldn't have that feel with that pistol. The only other pistol that comes close in feel for me is the Kahr K9, which, not so coincidentally, I also own, and I don't own many other highly-regarded pistols that don't have this feel.
Isn't it funny that decades before the term "ergonomics" became popular, Browning was creating ergonomically excellent pistols, while as late as the 1980s other designers were creating 2x4 feel in their creations?