I started back riding a bike around town last year, although I don't have a $600 mtb or road bike. I'm finding out a lot of interesting things that I took for granted when I first starting riding my old 10-speed back in the 70's, like tires and gearing. I was 15 years old when I bought my first 10-speed, comparing now to then, the bike has come a long, long way. I'm relearning everything I thought I knew about a bike and finding out that I don't like road bikes per say, reason being that I like off-road riding once in awhile and a skinny tire can't do that for me.
"Importance" is highly subjective, debatable, and relative. But, based on the limited information in your original post, and a lot of assumptions about you, I'd say: tire tread is only of moderate importance. Unless you're really pushing the limits of the tire (very fast aggressive cornering, etc.) you can make most common tire types (mountain, urban, cyclocross, road, etc.) work for commuting.
Some factors to consider:
- Commuting on mountain bike tires isn't usually super efficient, but can often be much more 'comfortable' than skinny road tires. This is because mountain bike tires tend to offer more cusion (due to higher air volumes and lower air pressures). Mountain bike tires often have higher rolling-resistance (when compared to narrower, higher-pressure tires) on pavement.
- There are circumstances in which mountain bike tires have less 'grip' on pavement. This is if/when the tires are especially knobby (big, spaced knobs have less surface area contacting the hard pavement).
- The trade-offs to these disadvantages is that mountain bike tires are generally better off-road, for those instances when you aren't on pavement.
Many commuters have found that a mid-width, semi-slick tire (i.e. something 30mm-45mm wide without significant knobs) ends up serving as a good/solid/comfortable on-pavement tire and also allows for sufficient off-road use.
Skinny is less common these days, even amongst the svelte roadies.
However for mostly tarmac riding with a chance of offroad you ideally want a tyre with medium sized side knobs and a flat surface for the crown (ie the bit that runs on the road surface during straight and level riding) Heres most of what I mean, but the knobs are quite large on this: