Sony intended the format as a replacement for the Compact Cassette. It provides a relatively low cost solution to making high quality recordings, and compared to Recordable CDs, it is smaller, cheaper, re-writable and allows non-linear editing. Compared to all forms of tape, it provides random access, meaning that you can quickly select songs with the touch of a button.
Although the format has been slow to catch on in the US, MiniDisc continues to thrive in Japan. Sales in the US were hurt by high prices and a format war with Philips Digital Compact Cassette (DCC). DCC has since been taken out of production (both Philips and Matsushita (DCC's Japanese backer) have now begun offering MD products). This fact, combined with the lower prices of current MD equipment, has helped MiniDisc begin catching on in Sony's US test markets.
The only drawbacks to MD are the fact that you cannot make a perfect digital copy of one (there is generational loss in every decompress/compress cycle) and the price of MD equipment (MD units costs at least US$200, though Sony's 1998 Bundle5 provides a home recorder and portable player for about $300). Blank prices have dropped 4 or 5 fold since the format's introduction and one can find 74' blanks in Japan for about $2 and in the US for $3 to $4. Blank prices are dropping in other countries too as MDs gain acceptance.
An unabashed Minidisc fan argues the MD case (vs. DAT). And it seems that people who actually sit down and spend a little time with MiniDisc, giving it an honest hearing with their own ears, become quite fond of the format.