What's the difference between "go on holiday" and "go for a holiday" ?
It is the difference between using the 'countable' form of 'holiday', and the 'uncountable' form.
In the countable form, a 'holiday' is specifically the entire trip - for example, a holiday to Disneyland. This is the 'go for a holiday [in Disneyland]' form.
In the uncountable form, 'holiday' is the time away. This is the 'go on holiday [for a few days]' form. The measure ('for a few days') is optional.
There is no real difference in the overall meaning of the two forms, though the first might be felt to slightly emphasise the fact that the holiday has a specific fixed length.
Of course, Americans can use the word 'vacation' in the same way that the British use 'holiday'.