Sorry, I see that is the beta map viewer in your screenshot now. This test does confirm that is how your map will look in the 4.x version of the JSAPI since it looks the same in the beta MV and the story map.
While all of the most popular layer types already are well supported in the 4.x JSAPI, there are still a few that have not fully been implemented. Over time the differences between the 3.x and 4.x JSAPI will be reduced as support for more layer types is added and enhanced. As those changes are made in the JSAPI, ArcGIS StoryMaps will immediately take advantage of them.
If you need the map to look the way it looks in the 3.x MV you can use the map to create a configurable app using one of the 3.x web mapping application templates (like Basic Viewer) and then embed that app in your story.