The Ted Lindsay (formerly Pearson) used to be awarded for "contribution to the sport of hockey" and changed definition in the late-1980s. It is also voted on prior to the end of the season, so it doesn't contain the full scope of the season (hence no Theodore nomination in 2002 and occasional flips compared to the Hart due to late-season races).
This is an important point, and is the single biggest reason why I think the Hart is more prestigious than the Lindsay/Pearson.
For example, in 2003, Markus Naslund was leading the league in goals and points late in the season. The Canucks were also leading their division (after Colorado had done so for eight straight years). When the Lindsay/Pearson voting took place in mid or late March, he was probably the right pick. Down the stretch, Naslund continued to play well, but Forsberg was unstoppable (18 points in the last 9 games). He captured the Art Ross very late in the season, and he helped his linemate Milan Hejduk win the Rocket Richard. Colorado also came back to win their division. Since the Hart is voted on after the season is over, it takes into account more relevant information - so Forsberg was the correct pick for the Hart. The Lindsay/Pearson is less prestigious because it didn't take relevant information into account.
Similarly, in 2006, Jaromir Jagr led the league in goals and points late in the season. Joe Thornton had an incredible finish (17 points in his last 7 games), which allowed him to win the Art Ross. He also helped his linemate Jonathan Cheechoo win the Rocket Richard (10 goals in those last 7 games). When the voting was done in mid/late March, Jagr was probably the right choice, but Thornton was probably the right choice taking into account the entire season.
2013 is a similar story. When voting took place in late March, Crosby was easily leading the league in scoring. He suffered a freak injury on March 30th and missed the rest of the season, allowing three players to pass or tie him in scoring. He was easily deserving of the Lindsay/Pearson when voting took place, but probably wasn't the right pick for the Hart since he ultimately missed a quarter of the season (which the players didn't know at the time).
There are other, similar examples (go back to 1972 for Jean Ratelle, who was having a historic season before getting injured late in the year - he still won the Lindsay/Pearson over Bobby Orr due to premature voting).
In summary - the Hart is the more accurate/reliable award because it's based on a full season's worth of information - the Lindsay/Pearson isn't.