Rent Allowance and Rent Supplement are completely different schemes.
Rent Allowance is an old Social Welfare Scheme and applies only to persons affected by the de-control of rents back in the 80s. There would be very few people in receipt of this, and no new applications.
Rent Supplement is part of the Supplementary Welfare Allowance scheme and is the supplement most discussed here and elsewhere - it's colloquially called "Rent Allowance" which leads to the confusion.
There are maximum rent limits set by Dept of Social & Family Affairs for each country, and for different family sizes. Rent Supplement is not payable to a tenant if their rent exceeds the appropriate limit (there are some exceptions but this is the general rule).
The amount granted depends on income (as it's a means-tested payment). The tenant must make a minimum contribution of €18 per week to the rent.
Rent Supplement may be paid directly to the landlord, although it's not compulsory (as it's the tenant's claim).