I suppose it indicates you're a more "well-rounded" person etc, but the extent to which things like the DoE are helpful is really exaggerated, particularly for very competitive courses. So no, doing the DoE probably won't in itself give you a meaningful advantage over someone without it.
I really would like to stress at this point that you shouldn't be doing things just for your personal statement. The fact that DoE won't help you get into Med school doesn't mean it's not worth doing. It sounds like a really great opportunity (not that I did it) and I definitely would encourage people to do it, just for the sake of the experience. I regret not doing it to be honest, it looks like a lot of fun.
I mean, I didn't really like the expedition and the only thing I liked was the people who I am now in contact with anyway. So the fact that silver costs £90 and it has little value is not very promising then.
Seen as you posted in the Medicine part of the forum I can only assume you are applying for Medicine. In which case it really doesn't make much difference one way or another.
Literally everyone in my year did DofE to bronze/silver at least, and I did not see the point in it. School helped them all the way through, and everyone was constantly like "it will look good on my personal statement"...?!