There is more than one right answer but the best answer is the one that is right for you. If you don't put anything: fish, water, filters etc from this tank into your new tank you cannot transfer any disease, but if you do then you run that risk. The problem in this tank may not be too serious and may come right. I would just keep the two systems seperate and see what happens. Neon tetra disease usually has a rapid course and is fatal so if that is not happening it will be something that could be curable.