difference between "," he said. " ." and " ," he said, " ."

“I was thinking,” she said, “that maybe you could teach me.” - (here after "said" if you put comma, in double quotation, the first letter should not be capitalized  )

“I was thinking,” she said. “That maybe you could teach me.” - (here after "said" if you put period, in double quotation, the first letter should be capitalized )

If the dialog tag is in the middle of a character speaking, then the dialog is not capitalized after the tag unless it starts a new sentence.  For example:

“I was thinking,” she said, “that maybe you could teach me.” [RIGHT]

“I was thinking,” she said, “That maybe you could teach me.” [WRONG]

“I love that dog,” she said. “He’s so cute.” [RIGHT]

“I love that dog,” she said, “he’s so cute.” [WRONG]

other similar concepts

Commas

Commas are always used with dialog tags, whether they come before or after the dialog.  For example:

“Look at the dog,” he said. [RIGHT]

He said, “Look at the dog.” [RIGHT]

“Look at the dog.” He said.  [WRONG]

He said. “Look at the dog.” [WRONG]

If an exclamation point or question mark is used and the dialog tag comes after the dialog, then there should be no comma and the dialog tag should not be capitalized.  For example:

“Did you see the dog?” he asked. [RIGHT]

“Did you see the dog?” He asked. [WRONG]