Service discovery vs load balancing
You should think about it as client side load balancing versus dedicated load balancing.
Client side load balancers include Baker Street (http://bakerstreet.io); SmartStack (http://nerds.airbnb.com/smartstack-service-discovery-cloud/); or Consul HA Proxy (https://hashicorp.com/blog/haproxy-with-consul.html).
Client side LBs use a service discovery component (Baker Street uses a stateless pub/sub service discovery mechanism; SmartStack uses ZooKeeper; Consul HA Proxy uses Consul) as part of their implementation, but they provide the health checking / end-to-end functionality you're probably looking for.
AWS ELB and Eureka differ at many points:
Edge Services vs Mid-tier Services
AWS ELB is a load balancing solution for edge services exposed to end-user web traffic. Eureka fills the need for mid-tier load balancing.
Mid-tier server refers to an application server that sits between the user's machine and the database server where the processing takes place in. The middle tier server performs the business logic.
While you can theoretically put your mid-tier services behind the AWS ELB, in EC2
Classic you expose them to the outside world and thereby losing all the usefulness of the AWS security groups.
Dedicated vs Client side Load Balancing
AWS ELB is also a traditional proxy-based load balancing solution whereas with Eureka it is different in that the load balancing happens at the instance/server/host level in a round robin fashion. The client instances know all the information about which servers they need to talk to.
If you are looking for a sticky user session (all requests from a user during the session are sent to the same instance) based load balancing which AWS now offers, Eureka does not offer a solution out of the box.
Load Balancer Outages
Another important aspect that differentiates proxy-based load balancing from load balancing using Eureka is that your application can be resilient to the outages of the load balancers since the information regarding the available servers is cached on the Eureka client.
This does require a small amount of memory but buys better resiliency. The Eureka client gets all the registry information at once and in subsequent requests to the Eureka server, it only receives the delta i.e the changes in the registry information rather than the whole registry information. Also, Eureka servers can operate in cluster mode where each peer is not affected by the performance of other peers.
Scale and convenience
Also, imagine, 1000s of microservices running and each having multiple instances. You will require 1000 ELBs, one for each of the microservice, or something like HAProxy that sits behind the ELB to make layer 7 decisions based on the hostname, etc. and then forward the traffic to a subset of instances. While with Eureka, you only play with the application name which is far less complicated.