"Changing and improving are not always the same thing" or "Changing and improving is not always the same thing"

If you use is, then you are implicitly treating the compound noun changing and improving as a single, grouped set. Which makes the sentence have the form X is not the same thing, which naturally leads to the question "X is not the same thing as WHAT?"

Clearly the intention is for you to compare changing to improving, so they are really two separate items; and therefore you need to use the plural are to make it clear that you are comparing them, individually, to each other.

Mismatches in grammatical number are tricky, and have no perfect solution that will meet with total agreement.

This though is not an example of that. Your sentence is of the structure:

[noun phrase] [third person simple present tense of to be] not always the same thing [elided "as each other"].

The third person simple present tense of to be will be either singular or plural depending on the grammatical number of the noun phrase.

The noun phrase "Changing and improving" is a plural noun phrase formed from two singular nouns joined by and.

So the correct form of the sentence is:

Changing and improving are not always the same thing.

We don't match the verb to the word before it, we match it to the phrase before it. The phrase is plural, and so must the verb be.