Good question, and the answer is complicated.
When you're forming a "habitual past" like the sentences here, there's not much difference in meaning. But "used to" can be used with any verb, and "would" can't be used with all verbs. (For instance, it doesn't work with love: "Duo would love me" doesn't mean that Duo loved me before. Instead, it sounds like the beginning of a conditional statement: "Duo would love me IF [something].") I don't know of a rule for when "would" doesn't work, so it's safer to use "used to."
"Used to" can also be used to emphasize that something is in the past. If somebody asked me, "who creates this report?" I might say, "John used to do it, but I don't know who does now." (I can't use "would" that way.)
"Would" is used for various kinds of conditional tenses ("I would eat if I were hungry" or "I would not do that if I were you" or "I would have told you if I had known"). I think this is probably how it's used most often.
Finally, "used to" can also mean "accustomed to." As in, "I am not used to the cold weather."
I'd like to add that English speakers use "used to" to introduce a narrative, while "would" is used after the narrative has already been introduced.
For example, if you and your friend walk by a restaurant, you would not start by saying, "I would eat there every week." However, you might say "I used to eat there every week. I would sit at a table by myself and eat fries for hours."
If it's an extended narrative, like a friend recalling his childhood, you'll here "would" over and over.
Sorry it's so complicated... : )
More to add:
As far as being able to use "would" for some verbs, but not for others, it's kind of the same as the rule about how iyor/ıyor gets translated.
If it's a stative verb like "love," seviyorum gets translated as "I love." But if it's a dynamic verb like "run," koşuyorum gets translated as "I am running."
Stative verbs cannot be used with "would," only with "used to."
Incidentally, as I was saying above, "used to" isn't used after a narrative has been introduced. So if we wanted to say "Duo beni severdi" in the middle of a narrative, we would probably just use the simple past tense and the word "then."
E.g., "I used to be a purple owl. Duo loved me then. He would kill me seven mice a night."
Again, sorry it's complicated. I hope you're getting the idea!
At first I thought you couldn't use "would" to make a past tense with a stative verb, but once I started thinking about examples, I realized it isn't as simple as that. For instance, "see" is stative. I can say either "I used to see ducks, when I went there" or "I would see ducks, when I went there."
I can use "would" with "be," too, in the past tense. "We met every day, that summer. I would go to the park, and Duo would be there, and we would sit and talk for hours."
I agree, though, that the use of "would" seems to become less predictable with stative verbs.
"See" and "be" can actually be either stative or dyanic (see here: http://www.perfect-english-grammar.com/stative-verbs.html )
However, I do think the way you used be, ("Duo would be there") was stative, but I think this example is irregular/colloquial, and an exception to the rule. I just hope there aren't so many exceptions that it's not much of a rule.
It sure is fun to think about all this!
I'm out of space to reply below, but I think there's an issue of "temporariness" at play. In the "Duo would be there" sentence, the "would" worked because the "be" referred to Duo's current location -- not what Duo was, but where he was at that moment. On the other hand, if I say something like "Duo would be young," no matter how I introduce the sentence, "would" can't mean "used to."
But when I go through the list of stative verbs, I come up with a lot of examples that work. In "I want pizza," want is stative. But: "Duo and I couldn't have dinner together. He would want pizza, and I would want Chinese, and we would always argue about it." I realized that even love sometimes works: "I used to make Duo cake for his birthday, and he would love it! But then, we found out he had a frosting allergy...." So on the whole, I think a stative verb rule would be very hard to apply.
(It's a little maddening, but you're right: it's fun to think about. I'm amazed by some of the strange things English does that I've always taken for granted! =) )