AISC's magazine, Modern Steel Construction periodically publishes a guide to specifying grades of steel to help engineers stay aware of developments in the market. Their latest edition, from this February mentions A1085, but still suggests A500 Grade C as the standard. They recommend checking that A1085 is available and affordable in your area - it sounds like you already have. This may not be true in all regions.
Other than the maximum yield strength which isn't useful in your practice, there are a few other convenient features. Firstly, according to this article, it has the same mechanical properties specified regardless of product form. By contrast, A500 Gr C tube has a different Fy for square/rectangular and round tubes. Also, A500 allows actual wall thickness to be reduced by a large variance. This was intended to allow for cheaper manufacture by less accurate methods, but with modern steel mills, tubes are consistently produced undersized. 296
Because of this, calculations require reducing the wall thickness by 7% from nominal. While these issues are not major considerations in specifying steel, they will have some appeal in simplifying the design process and reducing opportunities for error.
The standard also specifies minimum corner radii, which reduces the risk of corner cracks. Previously, AWS D1.1 and the AISC Manual have warned about the risk of corner cracks in square and rectangular A500 members subject to significant stress from welding or galvanizing. I'm not certain if this fear will go away, or just be easier to quantify if using the new standard since A500 is produced with fairly uniform radii already.
The main down side of specifying the A1085 tube is cost, as you point out. Additionally, you may not be able to count on availability right now if you are specifying projects outside of your specific area. One other down side if you do lighter work is that A1085 is not available with 1/8" wall thickness like A500 is.
In your situation, these considerations may be a wash, meaning you should continue to specify A500 to reduce costs as long as it is still available. It seems probable that the distinct advantages for seismic design will lead to widespread use of this new tube standard. If so, it will eventually become the default nationally, and you may have to switch simply because A500 becomes less readily available. In the mean time, it might be appropriate to allow tube steel in your designs to conform with either spec, since the design values are fairly similar. 296