There are, I feel, three main areas of the Franklin-Nashville Campaign which can be placed on Hood's shoulders as being a failing in his abilities as a commander, and they are all connected.
1) Spring Hill
Having decided that it was important that he catch the Army of the Ohio at Spring Hill and get between them and the Army of the Cumberland, General Hood decided to leave the battle in the hands of his subordinates - the very same subordinates he'd criticised after Altanta fell for giving a less than satisfactory performance - and had a meal and early night.
He removed himself entirely from the battle despite believing it was an important part of the campaign which would contribute much to its overall success or failure, and the Army of Tennessee failed to achieve any of his aims in his absence.
2) Franklin
Having failed to get between the Federals at Spring Hill, subsequently allowing Scofield to occupy Franklin, he had no choice but to attack if he hoped in any way to prevent the Federal Armies uniting. The wisdom of conducting a frontal assault on a wide front against an entrenched enemy will always be suspect - though there, in truth, may not have been many alternatives - and there is no doubt that it gutted the Army of Tennesse for no practical gain.
Having all but wiped out his officer corp, and suffered casualties numbering almost a third of his Army, General Hood felt he could do nothing but advance or risk his Army disolving though desertions, so that's what he did.
3) Nashville
With the Army of the Ohio united with the Army of the Cumberland, and both sitting in a major supply hub upon the Federal controled rivers, General Thomas and General Schofield were sitting pretty. Being almost double the size of their enemy, in a well fortified and well supplied position as winter set in they were never in any danger. Nevertheless, General Hood decided to attempt to besiege his enemy in the vague, forlorn hope that Confederate troops in the Trans-Mississippi would come to his aid.
Hood's failures were thus
1) voluntary abdication of command during a crucial battle upon which much of the campaign's potential for success hinged
2) willful disregard for the damage suffered to his own army and lacking the fortitude to admit the campaign was lost - some might think this harsh, but the Duke of Wellington once said the mark of a great commander was to know when to retreat and dare to do it, and Hood failed to do this after Franklin even though he must have known he hadn't really any chance of salvaging anything.
3) putting his army in an exposed position with no clear end goal, no hope of success, clinging to a fools hope of salvation and, in keeping with the above, being too proud to admit he'd failed, cut his losses and pull back.
I am of the opinion that Hood's entire career is coloured retrospectively through the lense of the Franklin-Nashville Campaign, that the utter disaster that Campaign turned out to be ruined his reputation forever more, and his good reputation as divison commander could not offset the damage.
This being said, the Confederates could have gained something from the campaign under a more competant Army General - Spring Hill could have been a victory that badly damaged the Army of the Ohio, it might not have prevented the Federals linking up but it could have depleated their numbers and undoubtedly would have left the Army of Tennessee in a far healthier state to face the Federals as it would have prevented the Battle of Franklin from ever happening - but I think any gains from this campaign were going to be undone in short order anyway, as the Confederacy was on it's last legs and would have been unable to exploit any success.