In writing, I have always believed I can use day numbers without the "th" at the end. I know they are pronounced, but have been taught I can write them either with or without this "th" at the end. Is this correct? And if so, is one of them considered to be better?
You can write either.
I will arrive in California on June 15.
I will arrive in California on June 15th.
There is no real difference in meaning between the two.
To refer to the date without the month, however, you need to add the "-th".
I will arrive in California on the 15th.
You also need the "-th" to invert the order:
I will arrive in California on the 15th of June.
Although I believe in British English it is acceptable to write:
I will arrive in Hereford on 15 June.
(British native speakers, please feel free to correct me on this. I am sure I have heard it and read this usage, but can't say for certain how common it is.)