Looking at the lineup of Fender and Squier I see that you can find standard and vintage guitars for about the same price. For example, Squier has both a Standard series and a Vintage Modified series. What does vintage mean here?
I get that it means that it is built the way they used to long time ago, but I see no big differences at first glance. Is it just the type of pickup they use or is there a larger difference between the two? Also, can I swap parts between them?
Over the years, Fender has refined the design of its guitars in ways which it feels make a better product for modern tastes. Modern bridges have more sustain, better resonance, more reliable and fluid tremolo mechanisms. Their current standard fretboard has a higher circumference - that is, it's flatter. Because of modern materials they can make the neck narrower, and hence easier to play. And they've got modern pickups made using modern construction techniques with modern materials.
However, some people don't want a modern guitar. They want a guitar that feels like a 1956 Strat, or a 1964 Telecaster, or whatever. Those with serious money buy a real used 1956 Strat in good condition. But for those who can't find one of those (but still have some money to spend!) Fender makes a Vintage range that imitates vintage models in all the important details -- everything from the design of the bridge to the spacing of the windings on the pickups is copied from the vintage original.
Whether or not this is something you would want, is a matter of taste.
A related market, is that of artist replica instruments -- where they borrow, say, Dave Gilmour's guitar, and copy not only its shape and components, but every scratch and scuff.
You hit the nail on the head. Vintage in this context means they are building the guitars a similar way they did "back in the day" but with some modifications. These modifications can be better woods, pickups, parts, etc.
I always thought vintage meant 15+ years old. Some say 25+, but it doesn't really matter. Vintage doesn't dictate value (as you can clearly see from these two guitars you are looking into).
I would guess that the Fender model probably has better parts/build but that is just from my experience. There are plenty of awesome Squier guitars though. Typically with most Fender guitars you should not have any issues swapping parts between the two. However, Fender to Squier, your best bet would be to ask a luthier or a guitar tech.
To add to the already adequate answers - ignoring hardware discussion, etc... to many it's about the sound of a guitar that will attract them to a vintage axe. A 60's/70's strat may get you that Hendrixian or otherwise classic rock tone. I know the 50's series strats were also sought after because of their unique tonality range, etc... your best bet is to take heed to things like sustain, etc.. that are related to hardware aspects but also the tone of the particular guitar. I'd recommend going in a playing them both to get an idea of the sound differences between them.
I once owned a Squier Vintage Modified instrument. Inexpensive, it looked like a brand new instrument but patterned after the look of the corresponding Fender model from many years ago. However it had the audio jack located on the side instead of in the front through the pickguard and had modern pickups and electronics.
Unfortunately the audio jack was defective and could not be removed by the repair tech at the store where I bought it. After experiencing the same problem with a replacement I moved on to another brand which was designed to be used and maintained but did not have the "looks".
The point being that with Squeir, vintage means looks, modified means "we did the improvements for you that musicians used to have a guitar tech do for them" and buyer beware it has no bearing on quality/serviceability.
A vintage guitar is an old guitar that is of exceptional quality.
I understand "vintage" to mean of or like a more perfect production of the past, as opposed to merely antique but not necessarily better.
