The Navy pays for nothing, it is the British taxpayer who picks up the tab - as with everything. There has always been a perception, probably exercised by both factions, of the differences between shipboard marines and the Commandos and the SBS.
The Commandos and the SBS being 'real' soldiers down among the dirt, with the big ship marines being responsible for bags of swank, big bands and bullshine. As for the description: "soldiers from the sea", during WW1 the marines role as light infantry was recognised in the title of Royal Marines Light Infantry and it was in this capacity that they maintained that role until early in WW2 when they were incorporated into some of the first Commando formations.
The eclipse of big ships from the British Navy meant that ship marines no longer had much of a home. That fact accompanied by the success of the Commandos in various 'brush wars' and post colonial skirmishes switched any remaining emphasis away from the bullshine bootnecks and ensured that a confident future lay ahead for the 'khaki' element of the Corp. If any element is required for service on the flat tops, I'm sure it will be those wearing the green beret and they'll serve with distinction much as they did on the former Commando carriers; Bulwark and Centaur.