Are there any advantages to having an axe with silk touch?

I have found a nice way to use the silk touch axe.

Melons when broken by any non - silk touch tool, or by hand, gives melon slices. However If you break melons by a silk touch tool it will drop melon block. So you can say no to the tedious melon crafting.

After you have a good amount of melons you can trade them for lot of emeralds.

Silk touch pickaxe is slower at breaking melons than silk touch axe So an axe with silk touch has its own unique advantages. This method of farming melons can be effective.

(P.S. melons can be harvested by silk touch tool is not mentioned in the Wiki)

Aside from the things mentioned in the other answers, a silk touch axe can also be used to mine bookshelves without breaking them. Breaking a bookshelf only drops a few books, and to craft those back into a bookshelf requires additional wood.

There are advantages of silk touch, coal ore, diamond ore, emerald ore, grass block*, huge mushrooms, ice, lapis lazuli ore, mycelium*, podzol*, packed ice, nether quartz ore and redstone ore are only optainable in vanilla survival through a silk touched tool. Additionally ender chests, bookshelves, glass and glass panes, can be retrieved after they are placed.

(all the items with the * sign on them can be also obtained by killing an enderman holding the item in his hand)

Source

The silk touch axe is a handy multi tool* for those with more emeralds than inventory space, along with melon, bookshelf and mushroom tree harvesting as previously mentioned.

*I use it when I want to harvest wood and silk in abandoned mines or wood and foliage (which are placed next to each other so not switching tools is nice) and don’t want to carry shears. It also has ok damage, so sometimes I forgo my sword as well, which gives me space for bow and arrows.