RAM is meant to be ever changing, so that data can be dropped when it is no longer needed by the program. ROM does not work that way. It is meant to store data indefinitely. Even when a ROM device allows rewriting the data, it cannot be read until the old data is erased and replaced.
Similar to ROMs are WORM (Write Once, Read Many) formats such as CD/DVD-R, and erasable ROM formats such as flash ROM and CD/DVD-RW. While these can be used as Mass Storage, especially erasable ROM, their primary difference from mass storage lies in the limited number of times each part of the media can be erased before it becomes too fatigued to reliably record new data.
For practical reasons, pure ROM chips are nowadays obsolete, having been replaced by PROMs (Programmable ROM, can be written only once), EPROMs (Erasable PROM, can be erased with UV light and rewritten), EEPROMs (Electrically Erasable PROM, which doesn't need a UV light to erase), and Flash Memory (an EEPROM that can be erased in blocks, hence it's suitable as removable storage).
Side note: Yes, write-only memory exists, as both a joke and as a real thing. As to the real thing, it's more properly "'write-only' only as far as the CPU can see"; there exists two versions of this: (1) the so called "bit bucket", or "/dev/null", where everything you write gets thrown away (useful mainly if you're dealing with some kind of mandatory output (e.g., progress reports when the program is supposed to be running in the background)) (2) memory that's actually an interface for hardware that the CPU can't read from, usually due to the associated manufacture costs; for example, imagine a sound device that plays a note depending on what bits are being sent to it, but is so cheaply made that it provides no feedback to the CPU.
Now, as for why some of you are probably at this page:
"Rom" (or ROM) is also used colloquially to refer to files full of data normally stored on ROMs. Typically this is either firmware (very simple low-level control software stored on a computer's board) or videogames, ripped whole from their original media. Generally speaking, "ROM" is used for data from games on cartridges. This is because there are existing formats for producing disc images from disc-based games, and the terminology reflects that, with people calling such things "ISOs" and the like. Many fans are really sensitive about this admittedly rather fine distinction, so never ever use, for example, the term "ROM" when referring to the digital images of ZX Spectrum games. There were only TEN games ever released on a cartridge for Spectrum, so all thousands of others are tape/disk images and nothing else — Speccy fans can be really picky about such details.
The legality of the distribution of ROM files is a somewhat controversial topic, as it relates to Digital Piracy issues, though it usually produces less backlash than the usual suspects (music and movies/TV shows). For what it's worth, the consensus is that it is illegal, though enforcement is not as prominent as for those other media, especially for older games, and such distribution may be part of the reason for the commercial success of such retro-gaming-related resources as the Wii's Virtual Console. Newer games, and especially computer games, are more subject to Digital Piracy prevention and enforcement means, but are often inherently more difficult to distribute (due to their larger size) and use (due to their hardware requirements) anyway.