This would be a very tough answer. Samurai were fast and precise, with all that armor a knight would have a tough time keeping up, and would tire out quickly. However, the katana is purely a slashing weapon, so I'm not sure they could get through the armor. Due to the knights basically having a sword and shield against a faster opponent who could find weaknesses in armor, I'd say the Samurai. The samurai are very precise and fast, I believe they would find weaknesses such as lightly armor areas such as joints or at least tire the knight out and be able to disarm him. The only real advantage I see the knight having is all the armor. While heavy, the samurai would have difficulty cutting through it, and if the samurai were permitted to use a naginata or similar spear I would say the samurai has a huge advantage. However, katana vs plate armor is a tough choice. I would just say the samurai would not get injured due to the slow movements of the well-armored knight and would at least tire the knight out enough to disarm and eventually kill him.
The show did mismatch a knight against an opponent with a gun, that was not a fair fight. However, just from what they covered of both groups (I have seen the knights and samurai, I love that show xD), I would say even that program they use would say the samurai would win. They are just too fast and precise for someone slowed by all that armor. Now, give the knight just some ordinary chain mail and a shield and sword, and the samurai using a katana with some chain/plate has little hope since chain mail is designed against slashing weapons. That's my opinion anyway, hope I gave some insight. :D
EDIT- I forgot to mention the training.. Duh.. Samurai train 24 hours a day and are ready to die in battle, this is defined in the Bushido. Knights do not train nearly as much and fight "not to lose". The Bushido states "If you fight not to lose, then surely you will. If you fight expecting to lose and trying to win, then you will win." (Not a direct quote but close..) The samurai would use martial arts and training to defeat the knight. I strongly believe practice and skill will overcome armor or fancy weapons.
To Gerard and Jurena - I do not underestimate the knights, they were deadly especially with their strength, however I just feel the knight's armor and lack of speed would not be enough to help them win. They can't kill what they can't hit, and although their armor is thick and the samurai cannot 100% get to them, the samurai can always find weaknesses in the armor such as weaker spots like joints, or the head. If the knight wears a helmet, then the vision is severely restricted and the samurai can at least bash him in the head until he gets knocked out. xD My point is they are too restricted by armor and the samurai is too precise and would be able to find the weaknesses. Please, if you can disprove this, do! I would like to learn.
The more modern encouters cannot be very reliable because when the Japanese did move further west, guns were emerging and there was no more technique. Again, please disprove me if you can but this question was about the 1450 knights and the 1600 samurai, so I feel speed and precision overcomes thick armor and heavy weapons.
The samurai mainly used horseback also, they were mounted archers, and even had weapons that were designed for taking enemies off horseback. Limiting them to 1 weapon each should keep the rest fair as well, at least give the samurai his horse as well. Even on a horse, with his right weapons the samurai could easily throw the knight off and take it to the ground.
Katherine - Did you know that the samurai were almost all mounted in their early eras? They also fought each other, so they knew how to take enemies off horseback very well. Limiting the weapons, just using the natural armor and most common weapon, I'd say giving the knight a horse and leaving the samurai with just a katana is unfair. Maybe a nodachi or naginata to even the odds. If the knight is on horseback then I would say it would probably be a short fight unless the samurai retreated, but again, that is not the fair fight matching two elite warrior classes while limiting their arsenal.
And again, I would like to say that using what the show covered on both warriors, I believe the samurai would win. The question was originated by the show "Deadliest Warrior" and based on that, Knight vs Pirate is not fair but I still was impressed with the knight. Samurai vs Vikings was a pretty good fight, and the samurai proved their worth. So, based on what the show covered of the two groups, and what I personally know and believe, I would say the samurai would win the majority of the time if it were a fair fight.
I would also like to say sorry for such a long post! It just takes more then a paragraph or two to get my point across. ^^ I agree you give the upper hand to the side you favor most, and I do favor samurai more I will admit that, but I like knights as well, in my opinion two of the best warrior classes ever.