A CableCard costs about $4.99/month.
I use Tivo (I have a Roamio although the newer Bolt platform also works -- you just need to make sure you get the version for Digital Cable and not the Over-the-Air version they also sell).
Upfront cost is the main issue. You need to buy the box plus pay for the guide service (guide service plus the cost of a cable card if you have only one TV and buy the guide service on month-to-month run about the same as a Verizon DVR box).
The savings come if you have more than one TV and use the Tivo remote boxes for those (links to one of the 6 tuners in the primary Tivo and you get all the TV and recording capability of the main box). You don't pay for guide service on these as they use the service from the main box.
The savings also come if you buy lifetime guide subscription for the main box and keep your Tivo for a while. Eventually you offset the guide cost and save money.
In my setup, I have a main Roamio unit and two additional TV's on remotes and the payback was about 23-25 months. Up to that point, you would essentially break even or be slightly ahead with the Verizon STB's. After the break even point however, the guide data is paid for and the box costs have likewise offset the Verizon month to month costs and everything after that is essentially the cost of the cable card monthly fee.
Note that this is covers box fees only -- the cost for the actual channel package (before you start layering on all the STB's fees) is still extra (and the same for either a Tivo setup or a Verizon STB setup).
Benefit of Tivo is you can copy shows off the box to your PC and if you ever change providers, you can take the box and any shows you have recorded with you to the new service provider. You need only swap out the cable card and relearn the channel package from the new provider.