This question is slightly different that the other referenced. In the referenced question, it was asked what the difference was when using count(*) and count(SomeColumnName), and SQLMenace's answer was spot on.
To address this question, essentially there is no difference in the result. Both count(*) and count('x') and say count(1) will return the same number. The difference is that when using " * " just like in a SELECT all columns are returned, then counted. When a constant is used (e.g. 'x' or 1) then a row with one column is returned and then counted. The performance difference would be seen when " * " returns many columns.
Update: The above statement about performance is probably not quite right as discussed in other answers, but does apply to subselect queries when using EXISTS and NOT EXISTS
