Noun and verb. Check, as in the mark one puts in a box. The only differences spotted so far are the new Applications icon in the left-hand source column (you’ll need to tick the box in the preferences before it appears). (Wired.com, July 10, 2008)/Our decisions in the voting booth should be among the most meaningful that we make in our lives. Too often, it seems to be just the opposite — we waltz into the booth and carelessly tick the box we’ve come to expect as our choice. ([Rutgers University] Daily Targum, March 23, 2011) Google Ngram. I normally don’t comment on my votes, but in this case I’ll say I “ticked” “Borderline” rather than “Over the Top” despite the fact that tick the box means the exact same thing as check the box and hence would seem to be pure pretentiousness. My reason is that the word check has so dizzyingly many meanings that I can see the utility of choosing tick instead.
notoneoffbritishisms.com 
I’ve come so late to this. But never mind. To a Brit, a tick is a mark (like a square-root/radical sign √ without the initial upstroke) showing that something is correct, while a cross (like an X) shows incorrectness. Do things differ in the US? In US auditors’ English I have come across the term “tickmark”, but it has meant no more than “checkmark” – ie any shape of mark, not just a tick. If a British form says “tick the box”, a tick is what is expected; contrast “(put a) cross (in) the box”. If he/she/it “ticks all the boxes” then all requirements are met. I think, in summary, that tick has a positive quality about it, and cross a negative quality, while check is a more neutral recent arrival.
I’m coming even later to this, but what you’re calling a tick is what we call a check or checkmark. Most Americans would understand “tick the box,” even if it sounds a bit foreign, but I don’t think would have a particular mark in mind. (“Tick” suggests to me a quick, single stroke.) And yes, the check mark generally is a positive versus a negative X, although we may be less consistent as an X could also indicate fulfilled versus an unmarked box, even if a form said to “check” all the boxes that apply.
I’m not sure you fully understand, Terry. What Brits call a tick is a specific mark, a short downstroke at an angle of about 4:30 on a clockface, leading to a much longer upstroke at about 1:30. It is very specific and never refers to any other mark. It is traditionally used in schools to mark answers as correct, with a cross for incorrect, and no variation in either of these. I believe this is the same in the US but without the name. In America it is just one checkmark (auditors: tickmark) out of many possibilities, with no specific name.
We brits don’t use “check” anything like as widely as you do. We don’t “check” our hats/coats/guns at the door, for example. When we finish a meal in a restaurant we are given the “bill”, though we may pay it by “cheque”. Generally, we ONLY use “check” for “verify” as in “I must check what time the train leaves”, or “It was the least I could do – I checked!”
I assume the american use of “check boxes” derived from “checking off” entries on a check-list – by ticking them! 🙂
Canadians definitely don’t “tick” off list items, we check ’em off. And we send “cheques” but “check” under the bed for monsters. The term “in a tick” doesn’t exist here. I’ve heard the rarer “turn it down a tick” in England, whereas the equivalent in Canada is tit/bit/little. “Tit” is my favourite because of its other connotation.
In the US, a checkmark is physically identical to the British “tick”. An American tick mark is usually a single stroke, like if you are counting votes (for example) and make 4 vertical strokes and one roughly horizontal 5th stroke through the 4. Checkmark (British tick) shape is the only common mark to indicate a correct answer, where an X or an angled stroke (American tick) could indicate an incorrect answer.