Performance: Select count(*) where... VS select column_name where
It really depends on a database you use while executing the query and gathered statistics.


Oracle DB.  If stats have already been gathered , the first query would return the amount of rows in 0.00001 sec. If have not been, then it will be faster in five more times than while executing the second query as it is retrieving large amount of rows instead of just one. 
For MS SQL will be almost the same thing. 
For MySQL or MariaDB it depends on the engine. But I assume these databases always try caching the value of amount of rows as the optimizer will do queries faster.


In addition, columns in WHERE clause and in COUNT() do matter as well. They could be indexes or not and etc. There are lot of criteria considering perfomance between the queries but in the most cases the first one is recommended for using to get amount of rows. I could tell you more precisely if I had certain example.