What is better(more green, less rain) in Mid March Ireland or Scotland?

It sounds like your true interest is Scotland, and if so then I'd say go for it. Weather is changeable in both Scotland and Ireland, no less in March than in other times of the year. You can be sure it will rain -- often sideways or "lashing rain" which makes an umbrella or poncho useless. It's entirely possible you'll have ample hours of warm sunshine too. There's likely to be snow on the higher elevations in Scotland, but at sea level not as likely. Bring layers of clothing, including boots, hat, and gloves, and apply waterproofing such as Scotchgard before leaving home.

You can examine historic weather patterns for various cities on the Weather Underground website. See, for example, https://www.wunderground.com/history/airport/EGPE/2017/02/10/DailyHistory.html?req_city=Inverness&req_statename=United%20Kingdom&reqdb.zip=00000&reqdb.magic=23&reqdb.wmo=03059

The Secret Scotland website gives good suggestions for realistic driving itineraries that will give you enough time to enjoy, not just rush from one destination to the next. https://www.secret-scotland.com/

If you are really keen to avoid rain, Nairn -- small city east of Inverness -- has the driest climate in Scotland.

there's no way on earth to second guess the weather in either Scotland or Ireland in a particular week in March.

the weather can change by the hour, you could be blessed with glorious sunshine, or persistent damp, grey weather

go where you want to go (which sounds like Scotland), wear suitable clothing. You're unlikely to see much snow except on the mountain tops.

http://www.bbc.co.uk/weather/2650225

Not that it counts for much but they are having a notably snow free year in the Highlands, there's hardly been any skiing available at Cairngorm and that's at 2,500 ft and upwards.

I think that Ireland and Scotland get a lot of the same weather. The North Atlantic is going to drive a lot of the weather. The gulf stream becomes the North Atlantic Current and swings by both countries. Sometimes the weather can come from the south, but not often. The west in both countries tends to be wetter. When I've gone walking (hiking) in Scotland, I've been based near Inverness. This allows us to go west if there is good weather, or try one of the other directions if the west is socked in. If you want to see Skye, I think it's best to bite the bullet, dare the weather and stay on Skye.

As for snow, it will be around in the mountains. Check out the traffic cameras though to see exactly what it looks like now. Do it when it's daylight there as they are live. :) Some will provide better views than others. It looks like the A9 is clear.

A number of sites group webcams for Scotland. Here's Rampant Scotland. Here's another. Note the ones for Glencoe are the Mountain not the pass. They are trying to get skiers to come!

Pam

Thank you so much guys for your suggestions! It was a great help. We are planning to go with Scotland and spending St Patrick’s in Belfast. This is what we have planned -

Visiting London nearby areas (bath and Cotsfield) and then flying to Edinburgh.
Take a car rental on Monday afternoon, heads to Stirling and then sleepover in Glencoe
Tuesday: Glencoe –Fort William-Glennfin Viaduct-Inverness (sleepover)
Wednesday: Attractions around Inverness
Thursday: Inverness –Perth-Edinburgh
Friday: Early morning flight to Belfast, rent a car and cover some of Northern Ireland Fri/Saturday
Sat Night flying back to London

Would appreciate suggestions about adding(greener places :P)/removing anything to our Scotland Itinerary.