Older lenses tend to have lower contrast and to be somewhat soft wide open. As the aperture is closed resolution increases, in the better cases to beyond the limits of what even a very good film can record. The low contrast is mainly for two reasons. First, coatings have improved, reducing veiling flare. Second, the lenses are not as well corrected as modern lenses, especially for spherical aberration. Older lenses often (though not always) render out-of focus information quite beautifully, and the low contrast can be helpful in getting good shadow detail. Wide open, they can have a "dreamy" look that many like.
Modern lenses are usually better corrected. They generally have higher contrast, especially at wide apertures, and higher resolution at wide apertures (f/4 and wider).
The modern Leica lenses are, generally, hyper-modern. They approach their optimal performance by f/2.8 or even by f/2, with extraordinary clarity and resolution even wide open. In technical terms, they are phenomenal. For taking pictures, many prefer the look of the modern lenses, and others prefer older lenses.
Probably the pinnacle of old-school lens design is the 50mm Leica Summicron, in the collapsible version or the rigid or dual-range versions. I have a 50 mm Summicron with the current optical formula (a modern lens -- yet the oldest design in the current Leica lineup, I think). In color, I prefer it to the older versions, but the older versions do look better for black and white work, in my opinion. The differences are not so large that I see a reason to trade, but some here have more than one 50. Some have several more than one...
Arguably the pinnacle of the modern school is the current 50mm Summilux ASPH. This is probably the highest-performing standard lens ever manufactured for 35mm format, at all apertures and over a range of subject distances.
If you search for these lenses there are whole threads, with example photos. There's a very long thread that compares the various versions of the 50mm Summicron (and some others), with many many examples,
here.