No cellular option, either (but it works over Wi-Fi)
You have to pay up to get the Series 3 with cellular, and pay an extra monthly cost for data usage with your carrier. But doing so allows the Series 3 to stay connected, use GPS for maps, make calls, stream music, get messages, and get information like weather and news. If you're considering any sort of cellular watch, the Series 1 won't help you. But: if you're on a recognized Wi-Fi network, the Series 1 will connect away from your iPhone. It's best used in places like a home or office.
Not swim-proof, but reasonably splash-proof
I can't swim with Series 1, but I don't swim often anyway. It's splash resistant (IPX7 rated), so it'll survive hand-washing, accidental showers with it on and maybe even a brief dunk. But, in comparison, the Series 3 can be used for swimming in salt or fresh water up to 50 meters.
It won't stream music over Wi-Fi
There's a small key difference between Series 1 and Series 3: Series 3 has some clever ways of streaming Apple Music over cellular or Wi-Fi, and also has a new Radio app that streams Beats One, curated stations, and a few radio stations (NPR, CBS Radio and ESPN). Apple Watch Series 1 doesn't have these. Instead, you'll only play music by pairing to your iPhone, or syncing playlists and albums onto the watch. Syncing music requires the Apple Watch to be charging, so it's not instant gratification. But WatchOS 4's improvements help the Series 1 feel more like an iPod, too, just like the Series 3.
Less onboard storage than Series 3 with cellular, but still enough
The Series 3 cellular model has 16GB total storage, which isn't all user-accessible. The Series 1 has 8GB. Realistically, you're looking at about 2GB of space for music storage. But that's more than enough for a basic collection of tracks to listen and work out with.
IonX glass is less scratch-friendly than sapphire
Apple Watch Series 1 only comes in aluminum, with too-easy-to-scratch curved IonX glass covering the display -- not the more scratch-resistant sapphire on step-up steel Apple Watch models like the steel one seen in the photos here. Keep in mind, though, the entry-level Series 3 models have the same aluminum-and-glass construction. I prefer the lighter-weight aluminum feel, but I've picked up some scratches on the screen over time. Not a ton, but it's annoying. The more scratch-resistant sapphire only comes in more expensive steel and ceramic models, and you'll be paying a lot for the privilege.
Battery doesn't last quite as long as Series 3 (without cellular)
One minor but noticeable difference is battery life. With WatchOS 4, Series 1 lasts me a full day and a little more. Series 3, with cellular turned off, lasted me about two full days on a charge. But, if you use cellular, expect that to drop down to a day of use at best. The Series 1, in that sense, splits the difference. However, in both cases, I charge the Apple Watch every day as a daily habit, so it doesn't change things much.
Not as fast as Series 3, but it's fine
The Series 3 dual-core processor has speed increases this year, and yes, that means speedier app-opening and a generally zippier performance. The Series 1's older dual processors are still fine, though. Would you notice the difference? Maybe. The Series 3 feels fast, and the Series 1 feels perfectly fine...but slower for doing multiple things like swapping watch faces, or loading fitness apps. That being said, if you don't care about GPS, cellular, or swimming with it, I'd go for the Series 1's budget proposition...if it's on sale.
Apple Watches are ephemeral things: they're not family heirlooms. Pick a model that will work for you now. In time, expect that it will, like all electronics, be outdated. I like spending as little as possible for a smartwatch. The Series 1 is that more budget pick. And it's fine.
The Series 3 is the better smartwatch, though, and at full retail I'd pick it over Series 1. But the Series 1 goes on sale a lot. And, a year later, it still works well.