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No, integration platform as a service and software as a service integration vendors are not the same. Software as a service (SaaS) is any software that's being hosted for you, not something you have to host on-premises or maintain hardware for. A SaaS integration vendor will help you set up that SaaS solution for your own needs.
An integration platform as a service (iPaaS) is a specific type of SaaS designed to connect multiple cloud services together. The idea with iPaaS is to help you integrate multiple solutions to create your unique set of services. Every company is unique, and no single tool will work for all companies. Some companies have on-premises specialized hardware, or require specific cloud-based workloads that need to integrate with other solutions.
One popular iPaaS that almost everyone can relate to is If-This-Then-That (IFTTT). IFTTT uses common programming logic to allow certain events triggered by one SaaS to cause a reaction in another cloud service. For example, your Nest thermostat could be triggered when you get home and could fire off a notification to your Facebook friends to let them know you're back. With other iPaaS solutions, you can connect more business-level logic; for example a trigger that causes the addition of a new contact in Salesforce to be added automatically to MailChimp.
Another popular iPaaS solution is CloudPipes. Self-described as "duct tape for the Internet," this solution integrates with many different SaaS solutions, such as hosted code repositories, customer contact solutions, search and news engines, payment solutions and more. With more solutions coming out, it's important to have something to tie everything together to make sure all of your cloud services are in sync.
Of course, all of these services have existed for a while now. It's not a new idea that you'll want some sort of "glue" to tie services into one another, but the term was used by MuleSoft to describe its new CloudHub project. The idea with CloudHub is that it becomes the complete solution to integrate multiple SaaS solutions with on-premises interfaces. A true iPaaS like CloudHub allows you to connect inside your internal network with on-premises solutions that may be behind a firewall.
The rundown on SaaS app integration
Explore some and old iPaaS vendors