Non-exclusive Tag means the max compensation is two 1st round picks.
With the exclusive Tag they could potentially get more than two 1st round picks...
While technically true, it's never happened and it's not really the difference between the two.
The Exclusive Tag means teams can't sign exclusive players to offer sheets. That doesn't necessarily mean someone can't be traded, the player just can't leave by signing an offer sheet and his new team sending back two first-round picks.
No one can negotiate with him except the team that tagged him.
With the Non-Exclusive Tag, another team can sign a franchise player to an offer sheet and the original team has a chance to match it. If the original team doesn't match the offer, the new team must give up two first-round picks as compensation in order to steal the franchise player away.
Any team can negotiate with and sign him.
Other differences:
The Non-Exclusive Tag is calculated by adding the franchise tag cost together from the five previous seasons and dividing it by the sum of the salary caps from the five previous seasons to get a percentage-of-cap number that is then multiplied by that year's cap.
The Exclusive Tag number is determined by the average of the five highest-paid players at the position, based on the upcoming season.
The Non-Exclusive Tag would have cost $26,895,000 in cap room.
The Exclusive Tag cost the Cowboys $31,409,000 in cap room.
Personally, I think Jerry and Stephen saw ghosts and misjudged the market for Dak and that Exclusive designation cost them an additional $4.5M in cap room. While some teams might jump at Dak in a completely open market, I just don't see which teams would have been willing to sign Dak to the $34 -$35M APY he wanted
AND fork over two first round picks. Even if I'm wrong and they did, the Cowboys would have had the choice of matching it, thus signing Dak to a long-term deal, or getting two first round picks. The way they played it, the Cowboys get to do this dance all over again next off-season, with Dak having even more leverage (approx. $38M 2nd year Franchise Tag) while dealing with a flattening 2021 cap. They potentially could lose him for nothing more than a 2022 3rd round compensatory pick. I think they completely misplayed this, especially since they paid Zeke and Jayon 2 years earlier than they had to. The focus should have been on Dak early and they had options (Zeke - 5th year option, Jaylon - restricted free agent tender) to utilize with those players. I think they just expected Dak to fold and give them a discount.
Time will tell.