Scalability is the ability of a system to handle the increased load on its current hardware and software resources. In a highly scalable system it is possible to increase the workload without increasing the resource capacity. Scalability supports any sudden surge in the demand/traffic with current set of resources.
Elasticity is the ability of a system to increase the workload by increasing the hardware/software resources dynamically. Highly elastic systems can handle the increased demand and traffic by dynamically commission and decommission resources. Elasticity is an important characteristic of Cloud Computing applications. Elasticity means how well your architecture is adaptable to workload in real time.
E.g. If in a system, one server can handle 100 users, 2 servers can handle 200 users and 10 servers can handle 1000 users. But in case for adding every X users, if you need 2X the amount of servers, then it is not a scalable design.
Let say, you have just one user login every hour on your site. Your one server can handle this load. But, if suddenly, 1000 users login at once, can your system quickly start new web servers on the fly to handle this load? Your design is elastic if it can handle such sudden increase in traffic so quickly.