I think the distinction to watch for is that sliding trays (drawers in a coin cabinet) will move the coins on their felt every single time the tray is pulled out, whereas lifted out trays in a box won't, unless the box itself is shook about. I've seen photos of coin surfaces being destroyed by wool felt, really bad damage, so its worth being aware of. However whatever is in abafil tray felt seems to do my coins no harm at all. And I doubt it would be possible to ever damage a worn bronze. FDC silver might be another matter. So, some caution needed but the nature of the trays felt and coins is important.
The material used in Abafil trays appears to be velvet (to the best of my knowledge made of cotton), while some other
types of trays are lined with felt (most appear to be wool). Felt will damage the coins, velvet will not. My coins are stored in Abafil trays, some for nearly fourteen years, and in examining them under magnification and comparing them to the original photos, I cannot detect any damage caused by this method of storage.
Andrew's point about lifting the trays rather than pulling them out is a
good one, but the coins will nonetheless shift and rotate over time just due to vibrations from the floor caused by walking in the room, opening and closing doors, etc. Even the process of moving the case containing the trays, placing it on a table, lifting the trays out of the case and placing them back inside, securing the latch to the lid of the case, etc. — however gently and carefully done — will cause the coins to move within their trays.
Still, my coins have not been harmed by the Abafil trays.
So I recommend Abafil trays. Stay away from anything lined with felt.