16 guage, sometimes referred to as "Sweet 16" is, indeed, a sweet gauge. With the correct shells, it's pretty hard to beat the pattern you can get from a 16 gauge. A true 16 ga. (one with a scaled receiver) can deliver just as hard a punch as a 12 gauge and still be lighter and quicker.
Having said that, it must be admitted that the gauge has not been as popular as either 12 or 20 for many years and, as a result, shells can be harder to find and tend to cost a bit more.
Still, they are far from impossible to find and the 16 gauge makes an excellent hunting gun. Clay sports are another matter - it would be cost prohibitive, IMO, to shoot clays with 16 gauge, unless you load your own.
Winchester's M1897 is one of the all-time classics and a near indestructable gun. The interrupted thread system employed in the take down mechanism is designed to adjust to take up any slack that develops. The gun doesn't shoot loose over time and actually improves as it's worn in (well beyond any break-in period). My admiration for this example of John Browning's genius is no secret. If it's a choice between the two, I'd put my money on the 1897 every time. As has been suggested, if you can get them both for a decent price, why not?