Are the words "toward" and "towards" synonymous? If not, when should I use one over the other? "Towards" usually sounds silly to my ear, but is that just me?
So this root has a rich history and has appeared in various altered forms in numerous Indo-European languages.
In German we have wärts which has the s: rückwärts (backward(s)) and vorwärts (forward(s)).
Evidently, in Old English the -ward root was either -weard or -weardes. So even in ancient times, there were already two forms: one with an es and one without. The idea that Brits use -wards, whereas -ward is a modern Americanism simply does not hold water since both versions trace back to respective Old English forms.
In any case, there is no need to have any qualms about putting the s on -ward or about leaving it off.
Indeed it seems that it is a British vs American difference. Note the ngrams below.
I agree with @carlo_R's comment and James Jiao's answer. However, towards isn't just limited to British English. For example, I grew up in the southern United States (Florida and Alabama to be exact) and say towards (my whole family does as well). Now, I live on the West coast United States and they generally say toward.
This is usually an American vs British English difference. Toward is chiefly American and towards is chiefly used in BE and most commonwealth countries. The former is foreign to my ears but obviously understood due to the proliferation of American media.