Are JPG and JPEG the same?
Yes. They are two spellings of the file extension for one image format, JPEG, named after the Joint Photographic Experts Group, the committee that wrote the standard. A .jpg file and a .jpeg file holding the same picture are the same bytes; only the name differs, and every program that opens one opens the other.
The format's official name is JPEG, and the standard behind it is published by the ITU as Recommendation T.81 and by ISO as ISO/IEC 10918-1. JPG is not a different or older version of it.
Why are there two spellings?
Because of old limits on file names. MS-DOS and early versions of Windows allowed file extensions of at most three characters, so .jpeg had to become .jpg, while other systems allowed longer extensions and used .jpeg. The three-letter form stuck, and most cameras and phones still name their photos .jpg.
You may also meet .jpe and .jfif. They are the same format again: JFIF is the name of the file layout most JPEGs use, and some programs name files after it. File4Edit treats .jpg, .jpeg, .jpe and .jfif alike.
Is one better quality than the other?
No. Quality is decided when the file is saved, by the compression setting the program used, and has nothing to do with the extension. Two .jpg files can look very different, and a .jpg and a .jpeg saved with the same settings are identical.
The same goes for .JPG in capitals, which is how many cameras name their photos. Windows ignores the case of file names, and so does a Mac unless its disk was set up otherwise, so photo.JPG and photo.jpg are the same kind of file.
What does change quality is saving again. JPEG is lossy, so each time a picture is opened, edited and saved as a JPEG, a little more detail is thrown away. Keep an original and export copies from it, rather than editing the same JPEG over and over.
How do you change .jpeg to .jpg?
Rename the file. On Windows, turn on File name extensions in File Explorer's View menu, then rename photo.jpeg to photo.jpg; on a Mac, edit the name in Finder and confirm the change when asked. Nothing inside the file changes, so there is no quality to lose.
For a whole folder, Batch rename files renames files to a pattern you type and packs them into a ZIP byte for byte, without re-encoding any of them.
A form that says it accepts JPG takes JPEG files too, whatever it calls them. If it checks the name strictly, renaming is all it takes.
What about converting to or from JPG?
Converting is a different thing from renaming: it changes the format, not the name. To keep a picture exactly from now on, the JPG to PNG converter writes a lossless copy; to make it lighter for a website, the JPG to WebP converter; to put photos into a document, the JPG to PDF converter. Each of them takes .jpg and .jpeg files alike.
And to change how big a JPEG is without changing its format, Compress image saves it again at a quality you choose, or finds the quality that fits a size you name.