Disconnecting internal devices from the internet when no connection is desired has the benefit that these devices are no longer externally accessible and thus attackable. While on the surfacet this seems "more safe", your devices are not really vulnerable unless they are either in direct communication with something or you forwarded some ports.
The risk to regular consumer devices from attacks over the internet is very small these days, assuming no risky behaviour from the user (e.g. downloading "free_M$_offic_[no_virus_100%_gurarantee].exe").
The downside is obviously you can't connect to your devices if you don't turn your modem on. This might now be a downside for you, if this is not a functionality you use.
I'll elaborate a bit on the points brought up in the comments:
What does it mean for a device to be in direct communication or have ports forwarded?
In a normal home network, your devices are not directly reachable from the internet. Someone externally can connect to your home router (or router/modem/wifi-access point hybrid), but not directly to your home computer.
The router uses something called NAT, which basically means that your home computer can connect to a remote server and that server can reply, but once that communication is done, the remote server can't reach your home computer anymore. That's a gross oversimplification, but it illustrates the concept that others can only reply to you - they can't initiate a connection to your home PC.
In order for them to be able to do that, you need to configure port forwarding, which basically gives everyone a way to talk to your home computer (as long as that is turned on as well).
This means that if your computer does not have any active connections and no port forwarding is configured, it's essentially not reachable from the internet.
What is the risk to regular consumer devices these days?
This would be a full question in it's own right, but I would wager that the average risk is rather small. A regular desktop PC, for instance, running a modern, up-to-date OS, will probably not have any publicly available exploits.
Even with other hardware you're likely to be safe, even if an attacker could remotely connect to them in some way. This includes smartphones, game consoles, smart watches, etc.
I'm not saying there is no way that these could have an exploit, because history has shown that there are plenty of exploits for modern operating systems. I'm saying the likelihood that an up-to-date product has a publicly available exploit is very low.
But what if the vendor does not care?
Usually, if a vendor really does not care, the security community likes to make them care. Imagine you buy a router from GreedyTech, who only cares about profit and not about security. Some security researcher sees that router, and knowing GreedyTech's reputation, he buys it to analyze it. A week later, he finds an exploit and - like a responsible security researcher - he contacts GreedyTech. GreedyTech is not interested in spending any money to secure their products, so they kindly tell him to go away.
The security researcher tries again, explaining at what high risk they put their customers. GreedyTech does not budge, and again tells the researcher in no uncertain terms to "go away".
The security researcher then publicizes the vulnerability, hoping that it becomes as publicly known as possible. Why so? The researcher does this in order to:
- Make customers of GreedyTech aware to buy a replacement as soon as possible.
- Gather attention from the media, hoping that the bad publicity will force GreedyTech to improve their product or suffer from a bad reputation (and thus less money).
So in a sense, even if a product is highly insecure, it'll probably not stay insecure for that long.