In Wonderware, the Industrial Application Server would be the equivalent of the gateway in Ignition.
InTouch viewer would be the client using Vision. InTouch design would be the Ignition Designer.
The Wonderware IDE would be the Ignition Designer also, mainly the SQL Tags part of it.
The Wonderware Historian is something I dealt with very rarely, mainly because it was fairly difficult to access from anything but Wonderware. I am not using the history in Ignition, as I am doing my own thing via gateway scripting and SQL Server, so I know very little about history in Ignition.
Wonderware has had three ways to do alarming that I am aware of. Originally it was SCADAlarm, which had essentially no ability to be configured programatically. That makes a big difference when you have hundreds of alarms going to several groups of people. They quit updating it years ago. Then they came out with an add-on for the IDE that had a couple of simple, yet very important things missing from it. We waited for months after requesting an update to it, but never even got a reply on our request. Now, as far as I know, they have gone to a third-party built software package that interfaces with Wonderware and is very similar to the Ignition Pipeline system. It looked pretty nice. But by the time it was shown to us, we had seen Ignition, and were pretty much sold on it immediately. Once IA gets a couple more things added to their alarming stuff, it is going to be everything we can ask for.
There are other parts of Wonderware that we never used, but the Industrial Application Server, the IDE, and InTouch are the main components that compare somewhat with the core Ignition software.
Hope this helps a bit.