I carry an Estwing hatchet and one of those saws with a kinda "D" shape to it but more pointed at the business end and with a rubber handle on the "taller" end. Most importantly, are they SHARP! I can sharpen the hatchet, but depending on how much use the saw gets, I just replace the 24" (?) blade on it when I think it is getting dull.
When camping I doubt you will be felling any standing trees unless it is allowed to drop the dead ones where you are headed for. Otherwise you are going to be finding ones that are down for one reason or another and you won't need to cut through a 2' in diameter log just to get a little firewood for the evening campfire.
Some parks do NOT want you transporting wood from one place to another that might have beetles or other infestations on them which is perfectly cool with me. I just stop along the way near my destination and off the highway, not on private land, you can usually find some downed wood that meets your criteria.....no larger diameter than maybe 6" and enough of it to cut for the nights you are going to be staying there.
So for me....a full size axe or a 3/4 size axe is overkill anyway and if I had to choose between one of them vs the saw and hatchet; the latter would win in a heartbeat. The saw for cutting to length and the hatchet to split some small stuff for kindling and medium sized wood to get the fire going.
Don't for get your SHOVEL....to put out any fire caused by an ember! I have a WWII US folding shovel that is good for many uses around camp for that and it can be used to clear the area around your fire ring from combustibles. Just remember to return the site to how you found it when you leave unless you are in a developed campsite!
If you are headed for the wet country...chop some kindling at home (8" x 1/2") and dip in paraffin to help get it going fast and cheat a little more with some newspaper kept in a zip-lock bag!