First, my compliments on your playing and singing. Very nice!
The two ukes both sound excellent. Definitely top-notch instruments. If I had to choose one, it would probably be the Collings. It's just a little richer and sweeter in tone. The Martin has a bit of that traditional Hawaiian jangle/brightness, which some people love but it's not my favorite. Your mileage may vary!
I am certainly not an expert but preferred the Martin, as the Collings seems a bit "cluttered". The Martin is well balanced and clear - its notes didn't compete with each other. The choice of what you played really demonstrated their differences. You're playing was excellent.
Thank you all,
In order to make the comparison more accurate, both ukes have the same strings.
To my ears, the Collings is a bit brighter and crisp, the Martin fuller, warmer.
The Collings is slightly more comfortable due to the radius fretboard.
The Martin sounds more mature.
Notice that we are talking about details, both are great.
If I had to choose only one, I think it would be the Martin : fuller tone, history, charism