The cheaper bike will probably require you to adjust the brakes and gears plus lube the chain every week or two, and a bike shop to replace the brake pads, chain and cassette every 3-20Mm. For you, that's probably twice in the winter, once in the summer. With rim brakes you will also need new rims/wheels every couple of years, more often if the roads are gritty. But a lot depends on your riding style and the enthusiasm you have for maintenance (hence the 3-20 range). If "lube the chain every week" actually means "the bike shop does that every month or two", they'll probably be putting on a new chain. Count a monthly service at ~$50 and the chain/cassette/pads as $200 and that's $1000/year in maintenance, or $600 if you do the monthly service yourself.
It's important to note that you can skimp on all of this and it just means slightly more expensive fixes and a slightly worse riding experience (gears that skip under power, more resistance from damaged bearings, cables or chains the break, brakes that don't really work). When I was a mechanic I saw bikes where basically everything needed to be replaced, but people were still riding them (and just wanted a puncture fixed or a new basket or something). The bad riding experience means an hour for your 18km commute instead of 40 minutes.
With hub gears and disk brakes, especially with a full chain case, you'll lube the chain every couple of months, put new brake pads in every year or two, and I expect that replacing tyres will become your main dirty maintenance task. With a single-cable hub (Shimano, SRAM) you'll have to adjust the gears after a couple of months (the free service) then again a few months before you need a new gear cable. If you run Marathons or some other puncture-resistant tyre you should wear the tyre out without getting a puncture (a puncture is often the thing that makes you look at the tyre and go "need a new one". I run a Marathon Plus on the rear and have only had one puncture in at least 20Mm. I did get a 3" nail through one though (with no puncture!)
Those numbers are why people often end up with a dedicated commuter bike that costs a fortune up front. New Rohloff Oil every year at $20 compared to a new Shimano hub every three years at $800 makes the $2000 up front cost of a Rohloff seem cheap (a Rohloff will last 100,000 kilometres or more... no-one really knows because there aren't many hubs that have done that distance yet, Rohloff only started about 10 years ago).