I'll just throw in my 2 cents worth here as I used both sites this weekend to book a flight in April 2016 to Lanzarote.
Times and prices were within 10 mins and 10 euro of each other so no difference there.
The difference in the two sites was pretty big - Ryanairs site was so much simpler to use, and faster. The Aerlingus site crashed twice and took a "one moment" please age to go from the initial selection.
Also, as one poster said, with Aerlingus when selecting/moving from month to month - the forwards arrow moves on the page so you have to follow it with the mouse. Don't they test these things?
Another big thing was, with Ryanair I selected my flights then then could select bags and seats. This meant that it was simple to select two passengers, and only one bag, which was what we wanted. On the Aerlingus site I had the choice of selecting a low fare, with no bags, or a flexi fare with bags. So it was all or nothing for the two passengers. I presume I could have done some jiggery-pokery to select no bags on that screen and add them in later, but I didn't feel like getting more frustrated.
To be honest, I would have given Ryanair the business on the basis of their website alone, in comparison to Aerlingus's one.
Also, on a separate note, their planes were newer and had more legroom than the Aerlingus one. (We just flew out with Ryanair and flew back with Aerlingus this month).