The horns serve to position your hand where it should be on the handle, and the lower horn acts as a brace against which the web of the hand can bear when starting, and you're supporting the weight of the saw to just stroke the tips of the teeth against the work. Once you're into the cut and relax your grip, the horns prevent the handgrip slipping out of the hand. Whether thy are really essential or not is moot - after all, they became gradually smaller and smaller as time went by, and finally disappeared altogether on modern hardpoint saws.
On 19th century hand and rip saws, there's a dip on the top of the handle that your thumb drops neatly into when you're sawing two-handed. The fingers wrap over the top of the handle forward of the grip. There's also, of course, the famous 'thumbhole rip' handle, specifically designed for two-handed use - arguably another non-essential gee-gaw to bling up a basic tool and sell it at a better price.
Pretty much everything else is decorative, I reckon. Can't think of any use for the lamb's tongue; indeed, it's only the fancier saws that have one. Bit like medallions - strictly bling, or a chance for the manufacturer or handle-maker to show off their artistic taste.
If that sounds snarky, it's not meant to be (well, maybe a little bit!). I very much like 19th century saw handles, not least because they seemed better sized to the average hand than many later ones.
