I switched from the built-in Admob support to the AdMob Extension and I had the same problem (in both versions). I investigated a bit and found out that the AdView that is responsible for displaying the Ad above the running game can get "destroyed" (cleaned up by the OS when the app is put into the background) and hence the ads won't be displayed anymore when the game is opened again.
I tinkered around a bit with the source files of the extension to create a quick temporary fix for this: I changed the "Initialize Admob banner block" in such way that it will look whether the AdView exists:
If not, the AdView will be created.
If yes, the already existing AdView will be made visible again.
This fix is not perfect yet: It won't automatically restore the ad that gets destroyed in the moment the player leaves the game, it will only allow the subsequent ads to show up again. Also please notice that if the AdView needs to be rebuilt because the player returns to the game, it might take a second or two longer until the ad appears on the screen.
If you want to try the fix with your game, please follow these two steps:
1. Locate the following files in your stencyl workspace folder:
stencylworks\engine-extensions\admob\MyAdmob.hx
stencylworks\engine-extensions\admob\dependencies\myadmob\src\com\abliblablobla\AdMob\MyAdmob.java
Replace these files with the files in the attached ZIP-File.
2. In your game, replace every occurence of the "Show Admob banner" block with the "Initialize AdMob-Banner" block (use the exact same parameters for this block as you have before) Don't worry, the initialization only takes place if really necessary, otherwise this block will act like the "Show"-Block did before.
Please use at your own risk and backup whatever files you deem necessary!
If this fix is working for you and enough people are interested in this, maybe we can ask Abliblablobla to review this and add it to the base version of the extension.