Have stayed at both, although it's been quite a few months since I've been at the Conrad.
Depends on what you want. Morrison is very much more a boutique property, with brand new rooms & services. Mostly excellent, but still a couple of occasional speedbumps. Conrad is a very mature, more corporate-y hotel. Rooms there could prob do with a refresh, although they are generally very very good.
Morrison is in a more buzzy area, and will be closer (although not much) to restaurants & the nightlife of Temple Bar just across the river. Conrad is closer to shopping and on a quieter street (at the weekend), just opposite the National Concert Hall, and across from St Stephen's Green Park.
The Morrison certainly will have a hipper and more cool vibe about it, and depending on your own family preferences, I'd probably suggest giving it a shot.
Oh, however - the Conrad has a car park directly under it, whereas the Morrison's one is a few yards down the street. Not a biggie, but fyi.... And Conrad is on the posh side of the city, where the Morrison backs onto a more gritty area too.
Hope this helps?
Thanks for your input, I'll weigh your pros and cons for both. ^