If your current pages can be accessed by http and by https, and if you don't have canonicals or redirects pointing everything to one version or the other, then one very significant "con" for that approach is that you are splitting your link equity. So, if the http page has 50 inbound links, and the https has another 50, you would do better to have one page with 100 inbound links.
Another difference is how browsers show/warn about non-secure pages. As well as any ranking factor they may associate with secure. Again, in favor of redirecting http to https. The visual handling can also impact conversion rates and bounce rates, which can in turn impact ranking.
As far as cons to redirecting, one would be that you might expect a temporary disruption to rankings. There will likely be a bit of a dip, short term. Another is that you will need to remove and then be careful about accidentally adding any non-secure resources (like images) on the https pages, which will then issue a warning to visitors as well as possibly impacting ranks. There is some consensus that redirects (and canonical links) do leak a very small amount of link equity for each hop they take. So, that's another "con". But my recent experiences doing this with two sites have been that after the temporary "dip" of a couple of months, if done properly, the "pros" outweigh the "cons".