Snow boots are usually best in lots of snow (which is not usually a problem in Iceland) but will be less safe and comfortable to walk in for longer periods of time or on uneven or icy ground. You can wear warm wool socks in hiking boots and then they will be just as warm.
10 Best Winter Hiking Boots of 2021-2022Make / ModelPriceInsulationColumbia Powderhouse Titanium$170600g ThinsulateLa Sportiva Nepal EVO GTX Mountaineering Boots$510GORE-TEX InsulatedScarpa Mont Blanc Pro GTX Boot$525GORE-TEX InsulatedLowa Alpine Expert GTX$450400g Primaloft•Jan 2, 2022
Plastic and leather soled shoes tend to be the worst for walking on slippery paths. Look for some snow boots or hiking boots that have a large tread, and soles with deep lugs and patterned for extra grip on slippery surfaces. Shoes with non slip rubber soles are particularly good for this.
You can wear your hiking boots in snow. A good winter hiking boot will accommodate the whole days' worth of conditions. It will be stiff with insulation and traction for hard-pack and icy conditions and be waterproof to fend of slush and water.