Yes both prophecies refer to the same figure with different names. Following is the evidence:
Role of different cultures and their languages
Azor Ahai and Prince that was promised apparently refer to same prophetic figure. They are just different names used by different cultures to refer to one supposed figure who would lead the men into Battle for the Dawn. People of Asshai call him Azor Ahai and People of Westeros call him Prince that was promised.
According to the World of Ice and Fire, In some other culture, legends point to the same figure with different names such as Hyrkoon the Hero, Yin Tar, Neferion, and Eldric Shadowchaser.
Melisandre's conversation with Maester Aemon
There is not much information available about TPTWP as compared to AA. By the description she gave for Azor Ahai's duty, Maester Aemon quickly knew that she was talking about TPTWP and Battle for the Dawn.
From ASOS, Chapter 78, Sam's POV:
“Swords alone cannot hold this darkness back. Only the light of the
Lord can do that. Make no mistake, good sers and valiant brothers, the
war we’ve come to fight is no petty squabble over lands and honors.
Ours is a war for life itself, and should we fail the world dies with
us.”
The officers did not know how to take that, Sam could see. Bowen Marsh
and Othell Yarwyck exchanged a doubtful look, Janos Slynt was fuming,
and Three-Finger Hobb looked as though he would sooner be back
chopping carrots.
But all of them seemed surprised to hear Maester Aemon murmur, “It
is the war for the dawn you speak of, my lady. But where is the prince
that was promised?”
“He stands before you,” Melisandre declared, “though you do not have
the eyes to see. Stannis Baratheon is Azor Ahai come again, the
warrior of fire. In him the prophecies are fulfilled. The red comet
blazed across the sky to herald his coming, and he bears Lightbringer,
the red sword of heroes.”
Notice that Aemon asked who was TPTWP. Mel replied that Stannis was Azor Ahai. Which means they must be the same person. Later Maester Aemon does not say that if they are different figures or not, he just asks to see Stannis's magical sword Light Bringer. He saw it and left. Afterwards when he was dying, He told Sam this, In AFFC Chapter 35:
The sword is wrong, she has to know that ... light without heat ... an
empty glamour ... the sword is wrong, and the false light can only
lead us deeper into darkness.
He did not say that Melisandre was wrong to consider AA and TPTWP to be same persons, He just says that she has the sword wrong. Which means that TPTWP must also posses the same sword and characteristics.
The Sword
By what we know of TPTWP, We know that he must be a Warrior, so he must need a sword. Back when Rhaegar used to think he was TPTWP, he said this to Master-at-arms of KL:
I will require a sword and armor. It seems I must be a warrior.
And we know AA forged his own magical sword to slay the darkness (By plunging the sword into heart of his beloved wife to temper it).
So Spake Martin
In GoT Season-2 DVD Extras GRRM also used the word in a similar way. He said:
He [Stannis] is discarding the gods that he has worshiped since his
childhood and accepting the red god and giving himself to the lord of
light. And in return, Melisandre sees that the lord of light gives
him a token of his role as the prince that was promised by an ancient
prophecy. And of course, that's his sword Lightbringer.
Notice that Mel usually refers to Stannis as Azor Ahai, but George states that Mel sees him as TPTWP. That should clear it all up.
You can view the talk here.
