I have not done research on Qtum yet, but I have been following Antshares/NEO. From my relative 'outsiders perspective' it does seem that even if Qtum and NEO are competitors, it is NEO that seems to be the frontrunner in terms of being wellknown. In the cryptocurrency industry, there seems to be a preferrence to 1 major player (Bitcoin vs all the other 'like bitcoin' coins, or Ethereum vs it's competitors). Whichever becomes #1 in it's relatively field, seems to be 'the one' people want. All the others who dabble in the same pond end up being mostly speculative as in 'It may go as high as Ethereum!', etc.
I'll definitely look into Qtum though. But, all things being equal, NEO seems to have more momentum and things going for it currently.