The Land Registry and Registry of Deeds are in effect constituent offices of the Property Registration Authority. They are mutually exclusive registration systems of title to land (not to the land itself - it is possible for there to be several different titles to the same piece of land e.g the freehold interest, and leases, sub-leases etc)
The difference between the systems is best explained by saying that the Registry of Deeds registers the existence of a deed, while the Land Registry registers its effect.
What this means is;
With a Land Registry title, the state has read the title and has determined that the person registered as owner is the legal owner of the property. It keeps the deeds as proof of what it has done. The Registry of Deeds merely says that a deed exists that claims to relate to the property, but if you want to see the effect of the deed you will have to read it yourself. The Registry of Deeds does not keep the deed, merely an edited copy of it, called a memorial, so you need to get that deed from the person who has custody of it.
Land Registry title is backed by a State guarantee, while Registry of Deeds is not.
To prove that you are the owner of property in Land Registry you only need to show a copy of the folio on which you are registered and a map shows the extent of the property, whereas with Registry of Deeds you need to provide the purchaser with the title deeds going back into history (to a good root) and allow the purchasers solicitor to read them
to see if you actually do own it, and you need to interpret the maps on the deeds to see what property you actually own. This can get very complex in urban areas, especially when the maps are not of great quality.
The history of the two Registries is a product of Irish history of colonisation and land agitation. The Registry of Deeds was created in 1708 by Queen Anne as a means of recording the legal priority of deeds which would have an effect in the result of a dispute over ownership, and is one of the oldest State institutions in the country. At the time there were very few land owners in the country, being primarily large landowners, so the system worked fine. The Land Registry came into existence in 1892 to as a result of the decades of land agitation which culminated in the Land Purchase Acts. With the vesting of land in small farmers from the landlords by the Land Commission, it was clear that the Registry of Deeds system would become so complex as to be useless, so a system devised in Australia in the 19th Century by Irishman Robert Torrens was adopted here to provide a simpler system of registration. This is basically the same system as is used today, and it is the system used in many countries throughout the world. Because the Land Registry was originally designed as a system for vesting agricultural land, the result is that much of the land in the old urban centres remains as Registy of Deeds title.
Government policy is to move all titles in the country to the Land Registry, and to eventually shut down the Registry of Deeds. To this end, fees for "first registration" (the process of converting a Registry of Deeds title to a Land Registry title) are low, and compulsory registration is gradually being extended across the country. At this point all land in the country is covered by compulsory registration except for land in Dublin and Cork, and they will eventually become compulsorily registerable. The problem is that given the nature of land, the process will take many years to achieve.