The Austrian, who was beaten by Nadal in straight sets in last year's championship match, pushed his illustrious opponent all the way in the first two sets, only to run out of steam.
Nadal clinched a record-extending 12th Roland Garros title with a 6-3, 5-7, 6-1, 6-1 victory.
"I was closer than last year in the final, especially in the first two sets," said fourth seed Thiem.
"So I'm on the right way. And I failed today, but my goal and my dream is still to win this tournament or to win a Grand Slam tournament.
"I will try my best next year again. I gave everything I had in these two weeks. That's all what I could do. Was not enough at the end, but I went very far."
The 25-year-old Thiem had claimed a stunning five-set semi-final win over world number one Novak Djokovic on Saturday, after poor weather forced their clash to be suspended during the third set on Friday night in Paris.
As a result, Thiem was playing for the fourth successive day in the final.
"Yesterday was one of my biggest victories in my career," added Thiem, who has now reached at least the last four every year since 2016.
"I failed to make the biggest dream of my tennis life come true, so I don't feel like I did yesterday.
"I felt fine physically, the first set was unbelievably intense, and the second set.
"Maybe I had a little drop in the third set, I don't know. But... against other players, it's not that dramatic, a little drop like that physically and also tennis-wise. But him, such a great champion, he uses the situation and goes all in."
Thiem has beaten Nadal four times on clay, but Sunday was the first time in four meetings he had managed to take a set off the Spaniard at Roland Garros.
He pointed to the 18-time Grand Slam champion's ability to mix up his game as a reason he is so formidable.
"If you ask all men in professional tennis, I mean, almost everybody will tell you that he's one of the best volleyers of our game. Because the last time he missed a volley was maybe seven years ago."
READ: Nadal tells Thiem: 'You'll win it one day'
Nadal, who moved within two major trophies of Roger Federer at the top of the all-time list, told Thiem after the match on court that he would win the title "one day" and said his vanquished opponent was a "great inspiration".
Thiem has been more consistent this season than in previous years, beating Federer to win his maiden Masters title at Indian Wells before knocking out Nadal en route to the Barcelona Open title.
"I'm very disappointed and sad that I missed my second opportunity to win a Grand Slam title, but it's okay. I mean, I'm used to it," he said.
"I lost my first two Masters 1000 finals, as well. It's obviously the most difficult thing in our sport to win one of the four majors."
The 'big three' of Nadal, Federer and Djokovic have now taken each of the last 10 Grand Slam singles titles between them since Stan Wawrinka won his third major tournament at the 2016 US Open.
"That's a unique and also brutal thing, I guess, in our sport, that I won six amazing matches," added Thiem.
"I beat yesterday one of the biggest legends of our game. Not even 24 hours later, I have to step on court against another amazing legend of our game, against the best clay-court player of all time.
"That also shows how difficult nowadays it is to win a Grand Slam."