The meanings are the same, but the tone is extremely different. "Neat" and "swell" are very old-fashioned, while "phat" and "fire" are modern Hip-Hop jargon. If you're not African-American, using Hip-Hop jargon can be potentially offensive to some people who consider it to be cultural appropriation. Also, you might not be understood by some people who aren't familiar with Hip-Hop terminology. I would suggest that the best informal word for expressing approval is "cool", but you could also use "awesome."

I think "cool" is by far the most common of these word in US English, and it is broadly used and understood across different regions and social classes. And it is a "real word:" Merriam-Webster calls it "informal," and not "slang." "Swell" hasn't been used since the 1950s. It's out of date. "Phat" is far less common than "cool." I don't use it and have never used it in my own life. Dictionaries call it "slang." I think of it as urban, and being close in meaning to "sexy." I have never heard "fire." I'm a senior citizen and this may be brand-new slang that I haven't heard yet.

This is what I hear the most: “That’s awesome!” “That’s cool!”