Update: I was unfortunately not aware of Shamus Young's severe criticism of Fallout 3 available here to link in the original piece and I regret that. It dovetails rather nicely with what I've written and it's much better executed than my piece. I strongly recommend anyone...
Each time you fully beat the game, your robe (either red or white) has a new pattern added to it. The first is just gold around the hem, but each time after that the pattern becomes larger and more intricate until it fully covers you. It's important to note that the game must be fully completed from beginning to end to unlock the next pattern. Skipping to a particularly chapter and then beating the game will give no reward.
Embroided Hem - Beat the game once.
Fifth Embroidery - Beat the game five times.
Fourth Embroidery - Beat the game four times.
Second Embroidery - Beat the game twice.
Third Embroidery - Beat the game three times.
After you've beaten the game once, pause the game and press select. You will be given the option to replay any of the chapters.
Chapter Select - Beat the game once.
Unlock the Hidden White Cloak
Complete the task prompted by the "Transcendence" trophy, which is to acquire all Scarf upgrades hidden in the game. You can do this over multiple playthroughs. Once you acquire the trophy, head to the start of the game (or just pause the game and select "Chapter Select". This option is unlocked after completing the game once). From the center of the area, look to the right and you will see three large cloths sticking out from the ground. Go over to them and simply walk into them. Your cloak will now be white and the scarf you start off with will be longer. Unlike the red scarf, the white scarf will automatically regenerate scarf power once you land after jumping.
Threshold: Cross the broken bridge without completely rebuilding it.
Mirage: Find the hidden desert flower.
Explore: Discover all cloth creatures in the desert.
Adventure: Pass through 15 gates while surfing through the sunken city.
Trials: Sneak through the underground passage without tearing your scarf.
Ancestors: Find a mysterious creature hidden in the temple.
History: Uncover all 10 ancient glyphs.
Crossing: Reach the summit with a companion and return to the beginning.
Rebirth: Finish the game and return to the beginning.
Companion: Finish the game with the same partner for the majority of the journey and return to the beginning.
Reflection: Sit and meditate with another player for more than 20 seconds
Wonder: Meet 10 or more unique travelers.
Return: Start the journey again after a week long break.
Transcendence: Collect all unique glowing symbols across one or more journeys.