I ship UPS daily at work, but both are going to be ok as it'll be Next Day Air either way. Firearms are supposed to go NDA. It boils down to a personal preference, and/or which one has a routing/shipping facility closest to you.
In my 6 years shipping parts across the country, to Canada, New Zealand, Chile and Uruguay (most common international locations) I've only had one package damaged and one package routed incorrectly (NDA to S Carolina, routed to Denmark lol). I ship between 55-65 packages a month, plus about that many inbound for just UPS alone.
There will always be a negligent driver that gets a box crushed, or there might be some poor taping and it comes open mid shipment. I prefer UPS, just because of how often I deal with them, plus I can generate labels from my desk and have anything picked up on a weekday.
Josh