Both engines have their strengths and weaknesses. The 7.3 probably has less overall weaknesses. The 6.0 biggest cons are the EGR system and head gaskets / factory torque to yeild bolts. The bolts quickly stretch causing head gaskets to fail and everything quickly goes down hill from there.
Mileage on the 6.0 will average you anywhere from 10 - 23 depending on how you drive, where you drive, tuner, mods, towing/hauling, etc. 7.3 will never see anything in the 20's even on a good day with a tail wind. But few 7.3's have head gasket/bolt problems either - at least not until you start boosting their power levels considerably.
The 7.3 is more the stock workhorse with few issues. The 6.0 is more the hot-rod that works and has issues. The 7.3 typically has less horse power & torque numbers - both stock or tuned - but dont let that fool you - its a hard running truck - in a good way!
Once you deal with the 6.0 issues you have a reliable, hard working truck that can return you decent mileage.
On the 6.0 do the full EGR delete (either buy a kit or do the DIY here on the forum - 6.0 Mod Help -> scroll down to EGR Delete) check your local state first though - make sure they dont test diesels. 2nd - call Beans Diesel and buy a ARP stud kit (should cost you around $400) and plan to spend a gook $2K in labor to have the factory bolts switched to studs. Might as well replace the head gaskets at this time - wouldnt hurt and would be a good maintenance item to do. There really is no "time limit or mileage limit" to do studs & gaskets. They are going to fail so you might as well replace it before it does. In the long run you will save a ton of money and headache to do the work before they actually fail.
Keep your oil changed - run a synthetic 5w-40. You will be fine.
Also - 6.0's like to work. Work them hard and they will treat you well.
Either way - enjoy your truck.
If you really want to go all out - swap in a 12V Cummins in place of either engine. Then you got a Fummins.
If you really are looking at 6.0's - stay away from the '03's. I have an early '03 and love it - but it was the first year of the 6.0 and as everyone knows - the first year is always the most problematic. There are a few benefits though - namely the EGR system though controlled by the ECM isnt tied into the code part of the ECM. So if you unplug it or delete it - the ECM wont throw a code indicating a problem. After & during '04 they started throwing codes on this issue.
The '05's started using coil front suspensions - all previous years had front leafs.
I think it was the '06's (someone correct me here) started using a heavier front axle - 6000lb in lieu of a 5400lb.
Good luck and dont be afraid to post questions on here. Great group of people that love our trucks!