user-1: 4... Nxb4 is never played on chesstempo database for +2200 players games, being Bxb4 the most popular move.

I don't find any significant change after capturing the pawn with the knight unless after 5. c3, black can play Na6 supporting an eventual c5 to block the a3-f8 diagonal. 
If 5...Nc6 it looks the same position as 4...Bxb4 5. c3 Bc5
Stockfish evaluates 4... Nxb4 as =:

Why doesn't this move appear on databases is what I am wondering.

user8291: After 4...Nxb4 5.Nxe5 looks like a very strong reply, gaining the important e5 Pawn almost for free and posing an immediate threat to f7.

user12873: In my database it actually appears in 131 games, or roughly 1.5%.
The move also isn't objectively bad. But it's unpopular partly because it unnecessarily limits Black's options. After 4...Nxb4 5.c3, Black is practically forced to play 5...Nc6, which transposes to 4...Bxb4 5.c3 Bc5 (as you mentioned). Meanwhile, after 4...Bxb4 5.c3, Black has many options (5...Ba5, 5...Be7, 5...Bd6, or 5...Bc5).
There's also the issue that 4...Nxb4 looks unnatural. The knight should generally stay on c6 where it guards the key centre squares e5 and d4. Finally, some people might think allowing 5.Nxe5 is much better for White, even though it actually gives Black a winning advantage after 5...Qf6!.

user19877: It comes down to flexibility. In most cases, Nb4 will end up in the same position, as you said, but Bb4 give you at least two more common alternatives, and a couple less common ones.
 [FEN ""]

 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 Nb4 5. c3 Nc6 (5...Na6?) 6. d4 exd4 7. cxd4

Compare this with this line where black can play Ba5, Be7, Bc5, and even Bd6 have been played a fair amount of times. It is, generally, favorable to make your opponent worry about more options.
 [FEN ""]

 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 Bb4 5. c3 


-1: <p>4... Nxb4 is never played on chesstempo database for +2200 players games, being Bxb4 the most popular move.</p>

<p><a href="https://i.stack.imgur.com/yPZ1W.png" rel="noreferrer"><img src="https://i.stack.imgur.com/yPZ1W.png" alt="enter image description here"></a></p>

<p>I don't find any significant change after capturing the pawn with the knight unless after 5. c3, black can play Na6 supporting an eventual c5 to block the a3-f8 diagonal. </p>

<p>If 5...Nc6 it looks the same position as 4...Bxb4 5. c3 Bc5</p>

<p>Stockfish evaluates 4... Nxb4 as =:</p>

<p><a href="https://i.stack.imgur.com/C0ubG.png" rel="noreferrer"><img src="https://i.stack.imgur.com/C0ubG.png" alt="enter image description here"></a></p>

<p>Why doesn't this move appear on databases is what I am wondering.</p>
 8291: <p>After 4...Nxb4 5.Nxe5 looks like a very strong reply, gaining the important e5 Pawn almost for free and posing an immediate threat to f7.</p>
 12873: <p>In my database it actually appears in 131 games, or roughly 1.5%.</p>

<p>The move also isn't objectively bad. But it's unpopular partly because it unnecessarily limits Black's options. After 4...Nxb4 5.c3, Black is practically forced to play 5...Nc6, which transposes to 4...Bxb4 5.c3 Bc5 (as you mentioned). Meanwhile, after 4...Bxb4 5.c3, Black has many options (5...Ba5, 5...Be7, 5...Bd6, or 5...Bc5).</p>

<p>There's also the issue that 4...Nxb4 looks unnatural. The knight should generally stay on c6 where it guards the key centre squares e5 and d4. Finally, some people might think allowing 5.Nxe5 is much better for White, even though it actually gives Black a winning advantage after 5...Qf6!.</p>
 19877: <p>It comes down to flexibility. In most cases, Nb4 will end up in the same position, as you said, but Bb4 give you at least two more common alternatives, and a couple less common ones.</p>

<pre><code> [FEN ""]

 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 Nb4 5. c3 Nc6 (5...Na6?) 6. d4 exd4 7. cxd4
</code></pre>

<p>Compare this with this line where black can play Ba5, Be7, Bc5, and even Bd6 have been played a fair amount of times. It is, generally, favorable to make your opponent worry about more options.</p>

<pre><code> [FEN ""]

 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 Bb4 5. c3 
</code></pre>
