Villages are delicious. Essentially a cantrip that gives you an extra action, which inherently combos well with any other action, even other villages! What's not to love?
Silvers are boring. Drawing it enables you to buy more. That is it.
Now let's get into it. Your initial draw: 3-4 (equivalent to 4-3). We'll ignore 2-5/5-2's for now, and analyze the first few turns.
You have two choices for your first two turns: buy
- a village and a 4
- or silver.
Now you have 12 cards - 7 copper, 3 vp, a 4, and either a silver or a village.
We will further simplify the 3rd turn by saying you drew the silver or the village - to allow for direct comparison. In other cases, it doesn't matter which one you have in your deck - you didn't draw it yet.
There are 792 total combinations, 330 of which contain the silver or the village. (5/12 of the time, in case you're wondering).
Let's look at the scenarios:
This will happen 35 times, or about 10% of the time. With the silver, that's a 6, which nets you gold. With the village, it will net you a card - 3/7 chance of getting you a copper, 1/7 chance of getting the 4, which you can use.
It depends on what you get with the 4, and it depends on what you can get with the 5, but this means, by itself, that with the silver, this will get you the gold 10% of the time, while for the village, it's ~1.4%, and only if the action is a +2 coin action or if it's a draw action which drew (at least) 2 copper. Since you do not have another action, the extra action is wasted.
3 Copper, 1 VP
This will happen 105 times, or slightly less than a third of the time. With the silver, you can purchase a 5. With the village, you have 4/7 chance of getting the copper, which gives you the 5, 1/7 chance of getting the 4, which you can use. Since you do not have another action, the extra action is wasted again.
This follows the 4 copper scenario, except for about obtaining 5's instead of 6's. Though it's worth noting that if your 4 is a smithy (draw 3 cards), there is a chance you might draw 3 copper - a ~10% chance. There is a ~3% chance of getting a 6 with the smithy purchased. This does not exist for the silver. Unfortunately, that means you'll have 1 or 2 coins next turn, which might or might not be desirable.
3 Copper, a 4
This will happen 35 times as well, which is about 10% of the time. For the village, this is probably a better proposition than the last, but only barely - your cantrip draw won't be as good, but otherwise most of the same analysis follows.
With a +2 coin action, there is a 4/7 chance you'll get the 6 needed for a gold. With a smithy, there's a ~10% chance that you'll get to 6, sacrificing the future (a little bit). With the 2, you can pick up a 5, and still use your action. No lack of support there.
Those are about half the possibilities. The other half is about the same - the general concept is that while village offers greater volatility in some circumstances while the silver offers more stability.
The math seems to be very close overall in yielding the most out of the 2nd cycle turns - the silver is the safer move, but the village offers (in the right set) longer term viability, with higher volatility overall.
An interesting observation is that beginners tend to favor villages while veteran tend to favor silver is actually optimal - with the assumption that players tend to get better, beginners might need that extra volatility to yield a better overall win probability. It might be part of what feels like "beginner's luck" in these games.