The two most obvious differences on 1804 Half Cents relate to the 4 of the date: on some coins the horizontal crossbar of the 4 ends with a vertical crosslet; on others the 4 has no crosslet. The Plain 4 is slightly more common than the Crosslet 4.
Underlying the Crosslet 4 are individual die varieties:
Cohen 5 (early state of the Spiked Chin)
All of the Spiked Chin varieties are also Crosslet 4's, but the Spiked Chin designation takes precedence.
Collectors should have no difficulty finding circulated examples of the Crosslet 4 Half Cent. Mint State versions are quite scarce, especially in grades above MS-63. Examples with original mint red are exceedingly scarce (most have toned to an olive- or chocolate-brown color) and none are known with full mint red.
Sources and/or recommended reading:
"American Half Cents (Second Edition)" by Roger S. Cohen, Jr.