The significant difference is that "any" means "one, no difference which", where as "each" means, "all of them, one by one". In the request to "tick" (indicate) the item[s] the user wants to purchase, "any" would perhaps not be limited to a single item, so the user could choose to "tick" all or several such items. The "each" indicates essentially a request to "tick" all of the items the user has in mind.
Tick some items you want to buy
makes a suggestion to the user to select any or all items, and there is no specific criteria given, so the user can use their own judgement (or just be random) in the choosing.
Same difference in the meaning (between "each" and "any") exists in the latter two sentences.
It is necessary to mention that "any" can be used with a singular (like in your examples) or with plural nouns. The use with a singular noun limits the choice to only one item, if plural is employed, the number of items is not limited to one; it can be one, none, or even all of them.