Engage has a wide range of more or less connected meanings. You can engage an employee (give him employment), engage a gear (make it active), or be engaged to marry (pledged), etc.
The underlying senses of engage involve commit, pledge, become involved with, attach, deriving from the rare/archaic/obsolete...
gage: Something of value deposited to ensure the performance of some action, and liable to forfeiture in case of non-performance; a pawn, pledge, security.
Deploy derives from Latin displicāre (to unfold). Originally this was only normally used of military forces, troops, in the sense of to open out so as to form a more extended front or line.
By lately, particularly with reference to mechanically-activated weapons systems, the two terms have both been increasingly used in the sense of employ, use. It's a bit fanciful to say they actually mean anything different in OP's context, but if Captain Kirk on the Starship Enterprise were to issue the orders...
1: "Engage photon torpedoes!"
2: "Deploy photon torpedoes!"
...the first could be interpreted as an instruction to prepare the torpedoes for use, and the second as an instruction to actually fire them. The words are used so loosely that in the above context, they're really synonyms - but if you ask "In what sequence would Captain Kirk give these orders?" (a loaded question, implying there must be a difference), I suspect the vast majority of native speakers would say "1 then 2".