I've read a few other answers on similar questions that talk about the lower contact angle and lower surface tension with soapy water (compared to tap water), but nobody seems to mention how this makes it more useful for cleaning (washing up, washing a car, etc) - is this because of surface tension or is it another physical property that I haven't considered?
The effect of soap on the surface tension isn't what makes it helpful for cleaning, but rather its hydrophobic-hydrophilic properties.
soap allows insoluble particles to become soluble in water, so they can then be rinsed away. For example: oil/fat is insoluble in water, but when a couple of drops of dish soap are added to the mixture, the oil/fat dissolves in the water. The insoluble oil/fat molecules become associated inside micelles, tiny spheres formed from soap molecules with polar hydrophilic (water-attracting) groups on the outside and encasing a lipophilic (fat-attracting) pocket, which shields the oil/fat molecules from the water making it soluble. Anything that is soluble will be washed away with the water.
Water has higher surface tension than the oil. Suppose when we add a water into oil. Water takes spherical shape. It not wrap the oil. Now suppose there is oil and I add sopy water. Due to lower surface tendency the water wrap the oil into it. Now the water has not much tendency for taking spherical shape. As sopy water wraps the oil we can give more water and oil will come out of clothes like a ball