I use both and determine which by weather, terrain and activity.  Admittedly I like donning my low cuts most of the time.  I would advise you get good “trekking or hiking” specific shoes.  Too often I see low cuts that are really around town walkers, and off trail they break down fast and loose support when wet.

I switch to my full height boots when the snow and muck are at there best, When I scrambling over unknown terrain, where banging my ankles is common, and when stability is effected by blind placement.

As noted by others weight of carry is a factor.  

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