The two services are totally different, yes both are offered to clear waste from a site but they are used for different purposes.
A skip hire service is used to store waste on site that is going to be generated over time, as opposed to a grab lorry which is used to collect waste that is mainly laid out on the floor and is in need of clearing away because it cannot be stored any other way.
Most of the time a grab lorry is used for very heavy and dense material such as bricks, muck, sand, concrete etc which would not be very efficient for a person/people to handle, the bucket on the grab lorries arm is used to speed up the process in 'grabbing' the waste to transfer it onto the vehicle.
Also the weight allocation is very different, an average builders skip can take up to 8 tonnes of waste before it becomes too heavy for it to be loaded onto the skip lorry whereas the grab lorry can take up to 16 tonnes of weight.
If you are in a position where you have a lot of heavier material that is being generated at a fast rate and needs clearing quickly then that is the best scenario for a grab lorry, alternatively grab lorries are also used for lighter waste streams if the waste has been spread out across a large space, for example in a fly tipping scenario, then it makes sense for a grab lorry to drive around picking up the waste as opposed to a team of labourers doing so.
The prices are very different and due to the size and the amount of waste a grab lorry can cater for. The grab option is typically more expensive.