Are These Songs the Same Song? (Little Dorrit, Downton Abbey)

Comments are very much appreciated! This has bothered me for months ever since I first watched Downton Abbey. I remembered when I first watched Downton Abbey: my mom and I were on the second episode and this song came on...





The first thing I thought was "Hey! That song is in Little Dorrit!" I could have sworn that it was the same song!

Watch from 0:00 to 0:44
I am asking for your opinion: are these two songs the same? They seem to have the same melody and the same musical instruments. Comments are welcome! 


 God Bless,