Its definitely a cardiac fitness thing, your heart will be so much healthier and you will build muscle tone where you used to have flab. So stronger muscles do weigh something too.
Go for it! And don't be embarrassed - everyone has to start somewhere! If anyone that you know should see you, they will be impressed that you've started / are running. I felt embarrassed too but all the comments I get/got when someone spotted me are positive. If you start now you could aim to run the Race for life in the spring comfortably, and that will be a tangible achievement that everyone will admire!
I was/still think of myself as a completely non-sporty type, have never set foot in a Gym, but I can run 5k in under 30 mins now. I did it bit by bit and have kept up the 3X5k per week for about a year now, even though it was not going too well in the summer (injuries..). At least now I'm back to running the 5 or 6k without any walking. I'm not very ambitious so I'm still happy to stick to my 5k runs rather than progress to longer distances. Longer runs also take more time, and the 40 mins out of the house are all I can squeeze in at the moment...