These two sound systems were originally developed for film, and there are major differences between film and home theater digital systems. Dolby Digital was the first on the block with Batman Returns in 1992, where it received rave reviews for being a much more precise and accurate reproduction of sound than analog systems. It was actually the second digital system, the first was Kodak Cinema Digital Sound released in 1990 with Dick Tracy. CDS was very expensive and had a tendancy to fail without any analog backup, and was relegated to history. DTS introduced a year later with Jurassic Park, and received more acclaim, due in large part to the excellent sound mix done for Jurassic Park. After this, Sony Dynamic Digital Sound was released with Last Action Hero, it is actually a 7.1 channel system, adding mid-right and mid-left channels. My personal ranking of the systems would go: SDDS, DTS, DD. However, the systems are so similar that the quality of the sound system in the theater and its acoustics have much more to do with the end result than the different systems. The main differences are technical and cost-related, generally concerning the studio and theater more than the viewer.
Ultimately, the systems are very similar, they are digital multichannel surround systems utilizing 5 or 7 full range channels and one low frequency effects channel. They all use compression algorithms to compress the data to between 18:1 and 4:1. They are allow redundant, allowing both an analog soundtrack and digital soundtrack to coexist on the same print. They all have approximately the same frequency range and dynamic response.
The main physical difference between DD, DTS, and SDDS is the location of the film data. Dolby Digital placed their information on the film between the sprocket holes, while SDDS placed copies on the both outside edges of the film, making it doubly redundant. DTS uses a timecode strip that runs next to the analog soundtrack which syncs to a set of CD-ROMs containing the digital information. DD uses a LED and CCD to read the data from the film while SDDS uses a laser and a photo sensor array. DTS uses standard CD-ROM drives and an led and photo sensor to read the timecode.
Because all three use different placements, it is possible for a soundtrack to be encoded all three ways on one print still retaining the analog soundtrack for backup. All three systems also can switch seamlessly from digital to analog and back if the digital track has been damaged. Because DTS prints only a timecode on the film, it is most resistant to wear as long as the discs are kept clean and scratch-free. Dolby is also fairly resistant. SDDS is most vulnerable, because of the higher density of data and risky location, but it uses both sides of the film and is thus ends up fairly safe.
The other difference between the systems is the sound compression algorithm. Dolby Digital uses AC-3. Dolby explains AC-3 (Audio Compression - Three) as a powerful noise reduction and perceptual audio compression algorithm. This is essentially true, AC-3 uses a process to lower noise on quiet channels (noise reduction) and also drops signal from close frequencies when one frequency is strong (perceptual audio compression).
SDDS uses the ATRAC (Adaptive TRansform Acoustic Coding) compression algorithm also used in the Minidisc. ATRAC is also a perceptual encoder, but also takes advantage of the fact that humans have difficulty discerning higher frequencies.
Finally, DTS uses apt-X100 in the theater, which is much less lossy than AC-3 and ATRAC. It does not attempt to do perceptive masking like AC-3 and ATRAC, but it does lower resolution (aka reduce noise) on silent frequencies. Because CD-ROMs have much more space available than film does, it is not as necessary to compress the data. apt-X100 is a waveform encoder, and does not take into psychoacoustic factors into consideration. For this reason, it drops less information, in other words, the rate of compression is lower.
Because DTS uses apt-X100, it is generally considered technically superior to the other two. However, most people cannot tell the difference between any of the digital systems except SDDS because of the extra two channels. Ultimately Dolby's advantage is its brand name recognition. DTS's theoradical superiority, combined with their willingness to work with studios to create a high-quality mix has secured its marketshare. SDDS's pair of channels is its edge, although it also makes installation more difficult. I would doubt there will be one winner in theaters for quite a while, as all systems are fairly equal. The next revolution will be full digital cinema, and the bandwidth required for video (around 45-100GB) makes uncompressed digital audio a drop in the pot.