This isn't exactly that kind of thing. This is like, I pick two coins off Ebay with a similar value, and ask you which one you would rather take if they were free.
I'd probably go with the double eagle. It's nice and flashy, maybe not an "AU63" but still plenty of luster. The half dime is beautiful, but I don't collect the series, and as a type coin I'd rather trade it for a common date in equivalent condition plus some other stuff.
The $20 Lib . . . I like both series very much, but it's so much harder to find such a mark-free example of a Double Eagle than it is a seated half dime.
Civil War era, high grade, CAC... at the right price, I'd get that half-dime: http://www.pcgscoinfacts.com/Coin/Detail/4380/64 Nice for a permanent collection. Maybe greater long-term upside. But maybe overpriced at $1500. And maybe a difficult resell if you need cash fast. OTOH, I think the gold is lower-risk -- easier resell. But it's the common "s"... if you pass now, should be pretty easy to find one later. I'd feel more comfortable with the gold. But dang, that's a nice half-dime! Tough call.
The Half Dime, 10 times out of 10. The Double Eagle is a poor strike from a common year, and the Half Dime is a high-end (PCGS has only 13 higher of over 100 graded) example of a popular and liquid overdate.
The half dine is a very nice coin and I do collect this series. However there are several whom Refuted the 1861/0 over date. If you study the over date lay overs you can see how they make their case. I still take the H-10
I am not surprised at all that more votes are for the DE. It is 1.) Gold. 2.) Big. 3.) Pretty. Both are sparkly, pale, dipped, and non-original. However, I generally collect a higher grade than 58. For that reason alone, I went for the half dime. Considerations that did not affect me at all, but which may (for some inexplicable reason) affect other buyers: 1.) PCGS vs. NGC. 2.) CAC vs. non-CAC