They were 2 very different regiments with different badges.
The one you show is a 1950s cap badge for the Royal Ulster Rifles. As their name suggests they recruited from the Ulster area and pre-partition were the RIR. They did not wear collar badges until after 1958 when they adopted them as a result of the 1958 mergers. There is no way what so ever that the Victorian soldier is a member of that battalion.
The collar badge shown is a pre 1901 collar badge for the Royal Irish Regt who recruited from the south of Ireland.