The main difference is that daryti is a completely transitive verb, while veikti is usually intransitive. That means daryti always requires a direct object: daryti maistą (make food), daryti pratimą (do an exercise). Veikti: bankas veikia darbo dienomis - the bank works on workdays (only a circumstance is specified to the action).
veikti is often used when asking what somebody is up to: ką veiki? (what are you doing? - what's up?). This can be said with daryti: ką darai? Here veikti has a direct object - ką, but this is the only direct object that is common for veikti that I can think of.
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