This has probably been asked a million times before, but I want to gauge people's opinions because this is really an opinion case. For picks and shovels, these enchantments give the tool vastly different purposes, but an axe works more or less the same on things like bookshelves, giant mushroom blocks, and melons with either fortune or silk.
A Silk Touch axe should drop all three as blocks, a Fortune axe should have no effect on a bookshelf and should give you more than the normal amount of mushrooms and melon slices. Is that no longer the case?
The only case where I think it MIGHT be fair to say it makes no difference is melons, since you can stick the slices obtained with Fortune III back together to almost as many melons as you started with.
Easy choice for me: Fortune. My axe, "Arborcide," is enchanted with Fortune III, Sharpness V, Efficiency V, Unbreaking III, and Mending. It's Fortune enchantment is very useful. Of course I use to cut down trees, but I also harvest potatoes with it. It makes a huge difference.
I decided I really liked Silk Touch on my shovel. Keeps gravel from dropping flint (I also have a Fortune III shovel called "Flint Digger") and keeps grass on dirt blocks. My Silk Touch shovel ("Silk Road") has been my everyday carry for a long time now. No way would I go back to a non-Silk Touch shovel.
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I've never wanted either on an axe because I don't find them to be that useful and the repair cost becomes extremely high (in 1.6.4 a renamed Efficiency V, Unbreaking III diamond axe costs 33 levels to fully repair with a new item, as low as 31 if the sacrifice is slightly damaged; if you add Fortune III the cost is now so high you must use a single diamond for 37 levels - times four for a full repair, meaning you also use 4 anvil uses and 4/3 more resources; not that any of this matters due to my playstyle but I'd rather put Fortune or Silk Touch on other items).
The same goes for shovels and even pickaxes, just Efficiency V and Unbreaking III (plus Mending in TMCW, which is the same as renaming in vanilla), with a few exceptions; Silk Touch on a pickaxe for mining Ender chests and emerald ore, Silk Touch on shears (which can get and require them prior to 1.9 if you want to collect cobwebs), and Fortune on some tool for harvesting potatoes (a hoe in TMCW, which makes high-tier hoes useful by making them required for Fortune to work on crops, while taking damage when breaking any block).
At one time I did use Fortune on my main pickaxe when caving but it does more harm than good given my playstyle; all the extra drops just fill up my inventory/Ender chest (or backpacks as I used back then) twice as fast (it is also a bit more useful in TMCW when branch-mining for my first resources due to an ore much rarer than diamond but I don't need it afterwards).
Which is better on an axe would depend on what you want to use it for.
For harvesting Melons, Mushroom blocks and Bookshelves, then Silk Touch. But it can be annoying when chopping down trees, since you'll get the leaf block rather than breaking it for saplings (or apples). On the other hand, it can also be useful, if less efficient, for grass blocks, glowstone, glass, etc.
However, a Silk Touch pickaxe could also be used for all of those things, and Silk Touch is arguably more useful on a pickaxe.
Fortune on an axe doesn't do much for its primary use. You'll get more saplings from leaf blocks, and, in theory, more apples (but the difference in apples is so small that's it's not really enough to add any weight to this discussion). For non-standard axe use, Fortune is great for harvesting crops and getting more glowstone dust. But again, a Fortune pickaxe could be used, and a pickaxe is a more practical and useful tool to enchant with Fortune.
So the truth is, with the exception of quickly harvesting large quantities of Melon or Mushroom blocks, neither enchant is particularly useful on an axe, and would be better applied to a pickaxe.
That being said, I have both in my oldest active SSP world, and I prefer the Fortune III enchanted axe, since I mainly use it to harvest my tree farm (I added Smite V to my Silk Touch axe and mainly use it in the nether for critical hit kills on pigmen).
What would be most useful on an axe is Looting, since an axe is a formidable weapon. But sadly, Mojang denied us that.