Why does my item name not show up in 1.13?

It's /give @p diamond_sword{display:{Name:""Sword of Power""}}

Use:

/give @p diamond_sword{display:{Name:"text":"Sword of Power"}}

or if you want red color:

/give @p diamond_sword{display:{Name:"text":"Sword of Power","color":"red"}}

In 1.14 and above, putting JSON text components into NBT string tags has been made a lot easier. Instead of having to escape every " with a , surround the entire NBT tag string with single quotes ' instead of double quotes ". This means you can freely include double quotes in your string without having to escape them. This will mean that you will need to escape every single quote though, but there will not be that many single quotes in your JSON text component unless you explicitly need them.

Note that a single-quoted string with the same text as a double quoted string count as aliases and are equivalent to each other. Single quoted strings are now the default for all NBT string tags that involve JSON text elements. This is evident because using /data get will show the Name tag as a single-quoted string, no matter whether it was set using a single or double quoted string.

Here is the 1.14 way to give yourself that sword:

/give @p diamond_sword{display:{Name:'{"text":"Sword of Power"}'}}

Notice the use of single quotes to surround the string in Name instead of using double quotes. This will make creating JSON text in NBT much easier!