The respective processes may be interpreted as:
checking the box -> there is a box, and mark it with a check mark if you want to select it (This is commonly used in hard copies [forms])
ticking the box -> there is a box, and click it to mark the same with a check mark (if you want to select it). (common in online forms/surveys etc.)
Moreover, "checking the box" is more flexible, since those who live in the era where forms are answered in hard copies can automatically interpret it as to selecting the option available. Ticking the box may not be that simple for a person who is not a technological savvy.
'Ticking' is similar to 'clicking' (hence, all the results caused by clicking is possible). 'checking' is not similar to clicking, but you need to click in order to place a check mark in online forms.
Hence, in online forms, the two are technically equivalent (per common sense/experience of computer users).