The animosity is not necessarily all private servers but it is very much there for horizon, there was either lots of over zealous supporters or an actual ad campaign resulting in horizon being injected all over the place on retail focused content. in addition to that you have ninja (the streamer) rumored to supply them with a large sum of money to run/develop their server.
back to the perceived ad campaign, responding to posts on retail video where the post says something like "oh wow ffxi is still around?" and the reply is "you should try horizonXI it is just like the old times". People perceive this as stealing a person who might have resubbed to retail, whether that is an accurate perspective of the situation who really knows.
In addition to all that there is the exploitation of server by the admin group and their friends, whether that be by leveraging their inside knowledge of bugs or upcoming changes or their authority as administrators, playing on the server you run obviously creates a conflict of interest.
And lastly a lot of people stream playing it rather than playing the actual game, so that is annoying.