In the summer a long blade, not machete, but golok or similar patter is much more versatile than an axe. The blade will readily cut small sapling, split with a baton if necessary and is much more effective than the axe in gathering boughs and lighter vegetation.
An axe will work better on thicker woods, but it takes a fair size to really slow done a long blade (4"-8"+ depending on type) and there is little need to cut that size wood for shelter, burning or general construction and in general it isn't efficient to do so and felling large wood is fairly problematic and dangerous.
In the winter when the wood is colder it gets harder to cut, and often needs to be split many times to be able to burn and thus an axe can be far more productive, and will also cut through thick ice far easier than a bolo style blade.
Larger axes are also very problematic to use one handed which is something to consider from a dependance point of view. If you get injured, especially with your dominant hand it can be near useless.
Many of the comparisons between axes and long blades are lopsided because often they are based on quality axes vs tactical fixed blades which are not actually intended to cut woods well. A more even comparison is to use log blades profiled like various HI khukuris, ABS bowies and Valiant goloks, all of which cut woods very well.
-Cliff