The 64GB phone will be faster in the same way that larger SSD drives are faster: more NAND cells working in parallel. Additionally, as you fill up your phone, the flash storage will slow down as it nears 100% capacity.
Translated: The 64 GB phone should be faster generally, and will remain proportionally less full, longer than a 16GB phone. (Which also helps with speed of starting/installing apps etc.)
If I'm reading this right, your friend is saying: "Your phone will slow down if you have more than 16GB of stuff on it"
You can rest assured that the 64 GB Phone will NOT slow down if filled past 16GB, but the 16GB phone filled to 16GB will become nearly unusable (very slow) at full capacity.
It mainly depends on the percentage of...