DSLR will shoot better in low light, at higher ISO levels, can shoot bursts of shots for longer periods of time, have more advanced tracking focus systems, can focus and shoot faster with less lag or delay, have shallower depth of field, typically more advanced user control buttons, can change lenses to suit different needs, have longer-lasting batteries, shoot in RAW mode for those who like to edit photos, have large sensors with bigger pixel sites, typically retain greater detail and allow more cropping room.
Ultrazoom cameras are lighter, more portable, have excellent focal range in a handheld package, more simplified interfaces, larger and more foregiving depth of field, good stabilization systems built in, allow for fast and normal live view using the LCD screen to frame a shot, often have lots of automated features built in for convenience or to aid lesser-skilled photographers, are much less expensive especially if lens costs are considered.
Generally, those are the things to consider. If you intend to get fairly serious about more than just daytime snapshots, a DSLR is the more flexible tool, but also requires a bit more dedication with the heavier bodies and often the need for multiple lenses. DSLRs can do 'simple' auto shooting too, but then become less of an advantage considering they're being used with lenses that don't really take advantage of the DSLR's ability and without user control to push to the camera's limits...and the 'auto' modes on DSLRs are usually considerably 'dummer' than those on P&S cameras, which have automatic scene recognition, face and smile detection, automatic image stacking, and more.
Some DSLRs have more of these features available to them than others - and those might be a good selling point - for example, some of Sony's DSLRs can allow fast shooting in live view with the LCD, have smile and face detection built in, smarter auto modes, built-in stabilization in the body, and tilting LCD screens.
Some of the hybrid mirrorless or fixed mirror cameras also might be a good middle ground for you - Sony NEX system, Olympus PEN system, Panasonic GH system, and Samsung NX system all have larger sensors like DSLRs, and interchangeable lenses, but shoot using live view & LCD screens, with P&S style interfaces and functions like intelligent auto, smile detection, etc.
It all depends on what your real needs are - you can be a good photographer, and a happy one, with all needs met with only a superzoom P&S - it doesn't mean you're any less of a photographer, just that you found the tool that does what you need it to do, and learn to use it to the best of its ability!
