“Just so. Now we will begin the dance. Remember, child, this is not the iron dance of Westeros we are learning, the knight’s dance, hacking and hammering, no. This is the bravo’s dance, the water dance, swift and sudden. All men are made of water, do you know this? When you pierce them, the water leaks out and they die.” He took a step backward, raised his own wooden blade. “Now you will try to strike me.”
Which is a direct reference to the myth of heavy and unweildy longsword.
Whether George Martin himself believes the myth or merely made Syrio misinformed, is anybody's guess.
Also, no, knight in full plate with a longsword is not certainly much less agile than someone with a rapier and no armor. He will be somewhat less agile... if the level of training is equal... but neither longsword nor armor are so heavy that they will significantly limit agility.
Main issue with full suit of plate is actually overheating and tiring out sooner due to added weight and insulation (and possibly restricted breathing depending on the helmet).