I'm not sure anyone has mentioned the biggest downside to local external HDD backups. If you leave it connected all the time, your backup will be gone in the event of a fire, theft or even a lightning strike or other power surge. Also, each time you connect the drive there is always the chance of a virus destroying the data on it if your PC is infected. You really need at least 2 drives that you rotate between, storing one in a fireproof safe or off site.
I think the most robust solution is to use both cloud backup (for continuous short term) home backup (for long term). The cloud service means you never have just one copy and the home backup means you aren't dependent upon a company sticking around. As for not having an Internet connection when you need one, does it really matter how fast you can restore your photo album?
I recently started using Dropbox with my first ever smartphone (and I'm an EE, go figure). I love the fact that it absorbs all the photos I take on my phone and they are automatically downloaded to my PC as well. When I sit down to spend the time every so often, I back them up on external HDD's or thumb drives so I have my "own" backup too.
We should consider ourselves lucky we live in an age where our memories can outlive theft, fire and quality degradation from aging unlike the days of slides or prints.