REST API -  Accessing objects in your organization using REST. You want to leverage the REST architecture to integrate with your organization. No WSDL requirement. Well-suited for browser-based applications, mobile apps, and highly-interactive social applications.

REST API provides a powerful, convenient, and simple REST-based Web services interface for interacting with Salesforce. Its advantages include ease of integration and development, and it’s an excellent choice of technology for use with mobile applications and Web projects. However, if you have a large number of records to process, you may wish to use Bulk API, which is based on REST principles and optimized for large sets of data.

SOAP API  - Integrating your organization’s data with other applications using SOAP. You have pre-existing middleware services that need to work with WSDLs and XML data. 

SOAP API provides a powerful, convenient, and simple SOAP-based Web services interface for interacting with Salesforce. You can useSOAP API to create, retrieve, update, or delete records. You can also use SOAP API to perform searches and much more. Use SOAP APIin any language that supports Web services.

For example, you can use SOAP API to integrate Salesforce with your organization’s ERP and finance systems, deliver real-time sales and support information to company portals, and populate critical business systems with customer information.