What are the pros and cons of designing an entire network using VLANs rather than design the network only using subnets?
There's no discrepancy between VLANs and IP subnets - they go hand in hand.
Organizing your network into different IP subnets enables you to provide a (security) structure and to limit the size of each broadcast domain (instead of having a single large one). Without VLANs, subnetting requires separate switches for each subnet and separate cabling for each interconnect between switches and routers.
VLANs enable you to use the same L1/L2 infrastructure - switches and interconnects - for all (some) VLANs. That way, you save on a lot of hardware and cabling and can handle changes very flexibly.