Neovim originated as a fork of Vim and while it continues to maintain backward-compatibility with Vim, the two projects have slightly different goals which has led them to evolve down slightly different paths. While Vim continues to be a great project with a great community, Neovim has added several new features that in our opinion significantly upgrade the user experience:
Lua
First and foremost, one of the major advantages of Neovim over Vim is the inclusion of Lua as a first-class alternative to Vimscript for plugins and configuration.
Lua is easy to learn, read, and write, executes quickly, and allows Neovim to benefit from a lot of great work being done by Lua's extensive community.
Tree-sitter
Tree-sitter is a fast document parser that maintains a syntax tree for each document as it is edited, which basically replaces slow and often-inaccurate regular expressions when implementing a variety of features.
Prior to Tree-sitter, syntax-highlighting, indentation, and folding were implemented using regular expressions, which was often slow, inaccurate, and lacked features such as the ability to handle nested code blocks.
By leveraging Tree-sitter's syntax tree, Neovim gains additional contextual information about the document that can be leveraged to provide accurate and consistent syntax highlighting, indentations, and folds, improved navigation between classes, functions, parameters, conditional statements, as well as some useful extensions to text objects.
You can learn more Neovim's Tree-sitter integration at the nvim-treesitter and nvim-treesitter-textobjects repos.
Language Server Protocol (LSP)
Neovim includes a built-in Language Server Protocol client, which provides a wide range of functionality. Whereas Tree-sitter improves the experience of working with documents, LSP provides similar benefits to projects, to provide improved code-completion, snippets, formatting, jump to definition, refactoring, etc.
Learn more about setting up Neovim's LSP at the nvim-lspconfig repo.
More Information
This is just a brief summary of the key improvements that Neovim offers vs Vim. If you are transitioning from Vim to Neovim you can find a complete list of the differences here