The captive Portal is indeed a little heavy on the esp82xx because it internally needs probably the web server component to work 
Please note that enabling this component will take up a lot of memory and may decrease stability, especially on ESP8266.
Web Server Component — ESPHome
To get a little more in-depth about memory usage (specially now that you have the 2 devices behaving differently) you may use the debug sensor which will inform about heap memory of your esps!
And is your yaml very complex/with a lot’s of compoents anyway or is it rather a simple one?