Hi Grant,
You don't actually mention what situation your going to be using either tool in, that obviously has great impact on what you choose. For instance are you working out of the back of a vehicle, or are you backpacking? What other cutting tools if any will you be carrying?
Also the environment you are going to in will have a big part to play in your choice.
And finally as several other guys have pointed out, your experience with axe and knife has a large part to play.
I myself carry a GB small forest axe, a folding pack saw, and a four inch fixed blade knife. This has always covered my needs in my environment, however this set up will not fit every environment adequately.
I find my most usefull tool in shelter building and fire wood gathering is the folding saw. My saw has an eleven inch blade and with this I have cut very large diameter wood. I have even cut and then used breaking methods to deal with even thicker limbs than a saw of this size could normaly deal with if just cutting was employed.
I love my GB axe. I use it for spliting firewood, shelter building, carving camp equipment such as blanks for spoons and pot stands etc and have even used it to garaolch a deer. Having said that I was taught by my Grandfather at a very young age how to use an axe safely without chopping off fingers or burying it into my own shin.

I would suggest to anyone thinking of getting an axe or hatchet for the first time to use it in the backyard for a while to get used to it, at least medical attention is easier to come by than when your out in the wilds. I believe GB axes still come with a very informative panphlet which gives good expanations of safe techniques.
I must admit I haven't really ever carried a large knife, it sounds stupid but one reason is that large knives seem to be frowned upon in the UK by sheeple yet if your found in the wilds of the highlands with an axe it seems more acceptable, (go figure). I can certainly see why some people are recommending a large knife due to their brush clearing abilities, but again for me this is not a problem due to the environment I usually find myself in.
Whatever you choose, as I have said before, get accquainted with the tool before you go off into the middle of nowhere.
Cheers,
Tony.