I typed up a whole big response but then the forum threw it away because it had links and I'm a newb. So I'll just say there's a big discussion about this on the iOS 9 subreddit, and another on the official Apple forums. It's not just you, it's a lot of us.
Some folks are giving the indication that phones on 9.02 can message each other without problem, while a phone on 9.02 has trouble with anything older.
I'm on 9.1pb4 and my wife is on 8.4 and this problem started last night around 7pm. Any iMessage I send her has a small chance of showing "Delivered" but most sit as blue bubbles for about 5m before flipping to Sent as Text Message. She's getting the iMessages immediately but my phone doesn't think so.
I tried all the fun stuff listed in this thread and the others. WiFi, LTE, reboots (for both of us), deleting the discussion thread, etc.
Now we sit patiently and wait for Apple to fix it.