The difference is that in a modal editor you have to explicitly switch to one mode to input text. In that mode pressing a key adds that key's character to the screen where the cursor is, just like a typewriter would. Then you have to explicitly switch back to another mode to do other things like move the cursor, search, select an area, or copy text.
A modeless editor always allows you to insert text and requires you to use a modifier key if you want a key to do something other than insert a letter, such as holding Control and pressing C to copy the selected text.
They are really not that different if you think of holding a modifier key as being in a different mode. The real difference is the explicit key strokes required to switch modes. In a modeless editor you can think of a modifier key as switching to one mode while it is being pressed and then switching back to the other mode when it is released.
This part of the accepted answer is totally wrong:
The primary appeal, originally a necessity in the early days of Unix computing prior to the widespread adoption of the mouse, is completely keyboard driven editing.
Keyboard-only editing works just fine without explicit mode switching. You can move the cursor and select areas of text with key combinations. For example, Ctrl+A to move to the beginning of a line, Ctrl+Spc to set a mark followed, then Ctrl+E to move to the end of the line thus selecting the whole line.