A finney is a ten-thousandth of a bitcoin.
So 1 finney = 0.00001 BTC
A satoshi is a 1/10^8 part of bitcoin.
So 1 satoshi = 0.00000001 BTC
For the purpose of computation and quoting of bitcoin in fraction, I think it is better to use finney as compared to satoshi. Because more the decimal you go the more complex and confusing it would be. So in that sense finney can smoothen it better whenever a fraction of BTC is being quoted.
Further, when the price of BTC will rise (like in 2017) the use of finney for fraction of BTC amount becomes even more profound. For example, a 10000 usd worth BTC will result in finney worth exactly 1 usd. So it is always comfortable to quote finney in that sense and satoshi will sound better only when BTC value will be like 100000 usd and when BTC value will worth 100000 usd, 1 satoshi will be exactly equal to 1 usd.