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Hi, I'm Mike G and I'll be assisting you. Tell me how you connected the new fan, how many wires and their colors in the box and the number and colors of the fan wires?
Your new fan is wired with a common feed to the light and fan motor thru a remote receiver. That makes it dependant on one switch and you'll not have independent control from the existing switches.
With the other fan being wired off 2 switches, the only thing you can do is install a universal remote in it. Put each fan on one switch and use them as an override to the remotes. That will give you operation but not together.
The problem is that the old fan is operated by 2 switches. One for the light and one for the fan. Putting the new fan on either will make it operate either with the light or fan with the old fan. You can get a universal remote for the old fan, replace the new fan that doesn't have a remote or get a second new fan. Those are your options.