Plastic is OK for short-term storage. Say about a month .... 2, 3, who's to say. If it produces good beer, then keep using it.
Plastic does let more oxygen in than glass, so long-term storage in plastic will allow more oxygen in to spoil the beer than a glass carboy.
Better bottles are supposed to be a superior type of plastic which lets in less oxygen than other fermenters.
Personally; for beers that go from the kettle to the bottle in a month or less, it doen't matter what they are fermented in as long as it's food grade.
For long term storage, it is a different matter. Doing a Lambic or Brett that will be in the carboy for a year or more, it might make a difference. From 'Wild Brews' (an excellent book by the way):
Large wooden barrel (Rodenbach) allows in about 0.75 ccs/L/year of Oxygen.
5 gallon glass carboy allows in 17 ccs/L/year of Oxygen (through a silicone stopper).
5 gallon HPDE bucket (I assume a sealed HPDE fermenter) allows 220 ccs/L/year of Oxygen.
Roughly 10 times the oxygen gets to the beer with plastic vs glass. Does it make a difference ........ that's the $64 million question. Me; I'm sticking with glass for anything I plan to age.
Someone commented that wood barrels let in a lot of oxygen. That's correct, but the barrels they use are very large (thousands of gallons), the the surface area to volume is very low, resulting in a low amount of oxygen compared to the overall volume.