The verb is 'to destroy'. 'Destruct' isn't really a word in its own right. It has the following very limited use: v.tr. To destroy intentionally (a rocket or missile) after launch. n. The intentional, usually remote-controlled destruction of a space vehicle, rocket, or missile after launching, as for defective performance or reasons of safety. (http://www.thefreedictionary.com/destruct) 'Distruct' exists as part of the noun 'destruction', the adjective 'destructive', and the verb to 'self-distruct' . This verb derives from the above meaning concerning rockets and missiles, but has also entered everyday language. So, unless you intend to be an aerospace technician, you can forget about the word 'destruct'.

The difference in meaning of the two as verbs is this: destroy - kill; end the existence of; ruin; destruct - end the existence of. So you can say: His life was destroyed by his parents' bitter divorce. You cannot say: His life was destructed by his parents' bitter divorce. The word "destruct" was coined by the Americans for no good reason at all. The meaning of "destruct" had always been covered by "destroyed" already. Please note that "destruct" cannot be used as an adjective. In "destruct button", it is a noun qualifying another noun. That does not make it an adjective.

The difference between the two verbs is that "destruct" can be used as an adjective. For example: If you push the destruct button, the building will explode. Destroy is used more often in English. For example it is better to say: The house was destroyed by fire.