Benefits in upgrading to a gold axe? Less swings to do the same job. Other than that? Not really. Gold pick at least is necessary if you get the meteor landing on your farm, and upgrading the pick in general leads to much faster clearing of the mines, which is always a good thing. But Axe up to Steel is all you really need. Then again, I'm also the guy who never upgrades his Hoe, and only upgrades his sprinkling can up to Copper. Sprinklers are simply more efficient.
The only benefit is for chopping wood faster. It also helps in the secret woods, for the stumps that give you hardwood, easier and faster to chop down with god ax (and iridium ax)
The main benefit is that a gold axe also has titanium grips, a teak handle, incised vine-enshrouded runes, and a personalized tribute from the famed blacksmith Bob the Vociferous. It can be sold in about 50 years for at least six times its current worth. As for everything else..well, ti's been covered.