Web apps are exclusively designed to be used from a web browser. These are combinations of server-side script (ASP .NET, PHP, etc.) and client-side script (HTML, JavaScript, etc.). The web browser (thin client) fetches the web server components installed at the backend infrastructure of the system for the heavy lifting in providing its core functional web services. Some common examples of web applications are Facebook, Medium, Twitter, Smashing Magazine, Flipkart, etc.
On the other hand, SaaS is an evolved web app. It’s equally used to access services over the internet like web apps but not always exclusively dependent on the web browser to work. SaaS products are built to give richer user customization experience and seamless integration with resilient and seamless back-end infrastructure. Some common examples are Evernote, Salesforce, Hubspot, Zoho, Dropbox. Etc.