This is on a somewhat layman's level. I'm not a security guru.
Looking at this in very, very broad and general terms:
Election fraud is definitely a serious issue that needs to guarded against pretty heavily, ideally by many third-party officials and monitors. If you have random people working at the polling stations, some may be tempted to throw away ballots or to fill out extras. You have a lot of random people, each being a potential, partisan security risk.
Therefore to reduce the risk, as well as to speed up counting, a given jurisdiction makes everything completely digitized (no paper ballots), and they fully automate the vote counting.
Just looking at this vaguely, some issues I see are:
- Genuine bugs in the software. Not all issues are malevolent.
- The organization that produced the software was indeed malevolent.
- Even if they weren't malevolent, completely outside hackers can still try to get in and interfere.
- Hackers from within the jurisdiction's staff and officials can also try to get in and interfere. Or even if they don't hack in the most literal terms, they can still find other ways to revise the final results after they've been produced, but before they've been presented to another party or the general public.
- The whole entire system works as an opaque, black box. This means trying to monitor the ballot collection, as well as the counting itself, has the same issues as trying to debug a software defect in a black box, third-party item. Logging information helps, but it is not the same as having a clear/white box.
Even if the software were developed by the jurisdiction itself internally, it's still just a small subset of that jurisdiction (which can still be corrupt) that would be immediately familiar with the code and how to analyze it for potential issues. The corruption issues and black box issue are still somewhat at play.
On the other hand, imagine another jurisdiction chooses to avoid computers entirely for the purposes of collecting ballots and counting the votes. This other jurisdiction still uses computers for things like verifying someone hasn't already voted or sending internal communications between staff, for obvious reasons. However the ballots are all paper ballots, they are all collected manually by hand, and the votes are counted - and aggregated together - by hand.
That means there is no hacking, and it also means that we are now dealing with something at least somewhat closer to a clear/white box. If you have corrupt individuals collecting the ballots and counting them by hand, you can also have security and monitors, both from within the jurisdiction and from third parties, watching them. And if they miss something in real time, video cameras can sometimes provide footage of an incident. And if both of those fail, you still have A set of physical ballots, envelopes, and anything else. It may not be The set that is genuine (ballots missing or added corruptly - or by innocent mistake), but having A set, heavily derived from the genuine set of votes cast, is often better than having none at all.
Altogether it is potentially much easier to monitor.
Now that said, the first jurisdiction may still very well be much more secure in its election process than the second, but this would depend on things like the resources invested in security, and more importantly, how well they each manage things.
However is the first jurisdiction inherently running an extra risk by relying on computers to collect the votes and/or to tally the votes? Is the first jurisdiction, compared with the second, doing the equivalent of using HTTP instead of HTTPS, writing data access code that blatantly omits SQL parameters, or leaving the car defrosting and unlocked for 10 minutes while they're getting ready in the morning?
UPDATE: A lot of good answers here. I think there were at least a couple that more or less tied for 1st place, so I would've liked to accept at least a couple of different ones.