Does variability of block solution time change as difficulty increases? We know that the expected average time to solve a block from any given point in time is 10 minutes when the network hash rate matches the current difficulty.

However, as difficulty increases are we going to see more or less variability, or does it stay the same?

ie If we compared a difficulty of 500,000,000 with 500,000, would we see more blocks taking over 60 minutes to be solved, or less blocks, or will there be no difference?
The variability is exactly the same no matter the difficulty. It is always an exponentially distributed random variable with mean 10 minutes (the mean could be slightly off if hashrate mismatches difficulty).

What we may see in the future however is a reduction in variance due to decrease of the block reward. As transaction fees become a more significant part of the reward, the profitability of mining will increase the more time has passed since the last block; just after a block is found, mining is unprofitable so the hardware may idle/switch to a different task. So the network hashrate will increase with time, meaning that few blocks will be solved quickly (hashrate is too low) and few will take a long time (the hashrate will be high so it won't last long), meaning less variance.