I was curious about this myself. It's all fine to read documentation and theoretical answers, but I like to balance those with empirical evidence.
I have a MySQL table (InnoDB) that has 5,607,997 records in it. The table is in my own private sandbox, so I know the contents are static and nobody else is using the server. I think this effectively removes all outside affects on performance. I have a table with an auto_increment Primary Key field (Id) that I know will never be null that I will use for my where clause test (WHERE Id IS NOT NULL).
The only other possible glitch I see in running tests is the cache. The first time a query is run will always be slower than subsequent queries that use the same indexes. I'll refer to that below as the cache Seeding call. Just to mix it up a little I ran it with a where clause I know will always evaluate to true regardless of any data (TRUE = TRUE).
That said here are my results:
QueryType
| w/o WHERE | where id is not null | where true=true
COUNT()
| 9 min 30.13 sec ++ | 6 min 16.68 sec ++ | 2 min 21.80 sec ++
| 6 min 13.34 sec | 1 min 36.02 sec | 2 min 0.11 sec
| 6 min 10.06 se | 1 min 33.47 sec | 1 min 50.54 sec
COUNT(Id)
| 5 min 59.87 sec | 1 min 34.47 sec | 2 min 3.96 sec
| 5 min 44.95 sec | 1 min 13.09 sec | 2 min 6.48 sec
COUNT(1)
| 6 min 49.64 sec | 2 min 0.80 sec | 2 min 11.64 sec
| 6 min 31.64 sec | 1 min 41.19 sec | 1 min 43.51 sec
++This is considered the cache Seeding call. It is expected to be slower than the rest.
I'd say the results speak for themselves. COUNT(Id) usually edges out the others. Adding a Where clause dramatically decreases the access time even if it's a clause you know will evaluate to true. The sweet spot appears to be COUNT(Id)... WHERE Id IS NOT NULL.
I would love to see other peoples' results, perhaps with smaller tables or with where clauses against different fields than the field you're counting. I'm sure there are other variations I haven't taken into account.