Assuming you want a lens for birding, then longer focal lengths will make it easier, so within your budget, the 170-500 is going to be more useful than the 70-300
There are of course 300mm lenses out there that will kick the 170-500 butt, even with a 2x TC fitted but guessing by your first choices, your budget doesnt stretch to those beasts
The 170-500 APO was decent enough if you stop down to f7.1 / f8, but of course this requires decent light. The 170-500 APO predates Sigmas HSM, so focusing is noisier and less sophisticated than Sigmas HSM long lenses.
If you could stretch your budget a bit, i'd go with the 150-500 with OS ( optical stabilisation ). I'm on my 2nd one of these, the first being when i used a Canon body, and now i use Nikon. Sigma also make the 50-500mm OS, but this is way more expensive, and probably nowhere near your budget.
Its not life or death having to have a lens with stabilisation, so perhaps you could try for the 50-500 NON OS Sigma.
Both the 50-500 / 150-500 should focus faster than the old 170-500.
If you would like to see some sample shots with the 150-500mm Sigma, let me know and i'll post a few.
Another possible is the Tamron 200-500mm, and although this doesnt use stabilisation, ( not a big deal really ) its probably optically in the same ball park as the 150-500.
Some people bemoan Sigma for their QC ( or lack of it ) so if at all possible, try them before buying, which i guess means you'll have to buy local from a shop, though you should still go for a used one, rather than a new one.
Both my 150-500 Sigmas have been fine so i dont have bad words to say about them.
