JPE is an extension used as an optional alternative for JPEG images. It is usually used by digital cameras. For example, some Sony cameras are able to save a high quality JPG image along with the low-res copy in JPE for sending via email.

HEIF is a container that can be a single image or a sequence of images which makes it ideal for Apple's Live Photos. It is based on HEVC (High Efficiency Video Coding). The format is more convenient than the old JPEG mostly because it is twice smaller despite the fact that is has excellent quality. Therefore you can storage more images on your Apple device which is great. HEIF was developed in 2013 by MPEG (Moving Picture Experts Group) but has become popular only when Apple decided to replace JPEG with it in 2017. However, not so many programs are able to open it which is why you might need to convert it to another format.