Welcome to Flyertalk and the BA Forum, andrewkeith5, it's good to see you here, and you aren't the first to find this area confusing, even some forum regulars find this area difficult to navigate. Welcome on board.

You can book with BA or EI and get fairly similar products.

If you book BA, and you book under a BA flight number, so a codeshare or "sold as" flight, you should get some Avios and Tier Points. If bought via aerlingus.com you wouldn't get Avios and Tier Points at the moment, but you could get Avios via their new AerClub (personally I'd leave that a few months for it to settle down). No lounge access for BAEC members, but there is for AerClub status members at LGW due to the use of a third party lounge - well that is almost certainly going to change in the new year, or so I'm told. There are paid for alternatives at LGW, including within aerlingus.com

The checked baggage is fairly strictly enforced on EI, no matter how you do, and yes under BA it's one bag at 20 kgs, under EI there are different options. EI makes a lot of money from baggage fees, which is why some basic tickets are very cheap indeed.

If you book under BA you're unlikely to be able to get seat selection very easily, personally I wouldn't bother, I just sort it out at the airport and LGW's handling agents for EI are helpful in my experience. You should be able to check-in online via BA.com, and you get one shot at getting a boarding pass via there. without seat selection. But you may be able to get a boarding pass and/or do seat changes via aerlingus.com or the App, though it may only be in the final 24 hours. Aer Lingus has a very clunky IT system that manages to make BA look slick. No priority boarding on EI if booked via BA, but I don't presonally see the value of that on a very short flight.

You do get this trio of benefits if booked via Aer Lingus instead of BA, you lose the Avios/TPs, and you gain better baggage booking options, at least in online form.

If you go to the forum Dashboard you'll find a thread on travelling to SNN/ORK/NOC which while not directly relevant to your travels has a bit of Aer Lingus background information in it.