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

Looking at the page from the Minecraft Wiki, it has a few uses, which are normally unavailable to non-silk-touch axes:

• Harvesting Huge Mushrooms
• Bookshelves
• Harvesting Ice
• Mycelium
• Podzol
• Packed Ice

I bolded the huge mushrooms because it is the only item that it is truly the fastest to harvest with an axe. All the other blocks in the list are better to harvest with another tool, but I put them in because it is possible to harvest with an axe.

(By the way, the axe works like a normal diamond axe, with the added ability to harvest the above items)

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.