Here are some weather charts. The choice is yours.

http://weatherspark.com/averages/28818/9/Dublin-Leinster-Ireland

http://www.weather.com/weather/wxclimatology/monthly/graph/EIXX0014

>>is there any significant difference between these two time periods? <<

Absolutely no way to know until the 15th of Sept. Then you might have an idea which two weeks would be nicer. The difference could be this much ||, or no difference at all, or the beginning of the month could be better, or the end of the month could be better.

And historical averages won't tell you anything - other than what happens on average

Go when it is most convenient for you . . .

great, thanks guys.

The only other factor is light - days do start getting shorter the second half of the month, so you will have less daylight to sightsee. However, prices for airfare will likely be lower, as well.

In September 2009 we spent 5 days in Ireland from September 16-21 and the weather was wonderful. In September 2010 we arrived on September 28th and spent 8 days visiting and the weather was much more rainy and colder. As janisj said you really can not be sure of the weather but you do have a bigger chance of cooler weather at the end of September, from my experience. Enjoy your trip, I hope you will love Ireland as much as we did.

Makes No Difference at All.. Not one of the past 10 years have been the same other than the sun shines, it rains and the sun shines again. It can be windy, Ireland is first hit for the Atlantic after all. Sunrise to Sunset is 2 hours shorter at the end of the month than the beginning but 8am to 7pm daylight (of varying degrees) throughout the month so unless you aim to be on the road after 7pm and before 7,30am then you wont notice the change.
Personally we will as usual be in Clifden for Arts Week at some point between 19th-29th and other local trips will be based on festival/event dates rather than sunrise/set times.

"but you do have a bigger chance of cooler weather at the end of September, from my experience."

Experience over two visits is merely anecdotal. It honestly makes no difference. Or more accurately, it might make a difference, but what that difference is you can't possibly know until next Sept.

janisj Just wanted to give my experience as an example. I certainly am not a local nor an expert. We definitely have learned Irish weather is ever changing.

hdavig, you will enjoy your visit to Ireland no matter what the weather will hold for you.