First, DSPS is a chronic condition. If untreated, it causes "insomnia" virtually every night. If you only occasionally have difficulty falling asleep (e.g. a few times a month), DSPS is not the cause.
The most striking difference between DSPS and chronic sleep-onset insomnia is that that people with DSPS have at least a normal - and often much greater than normal - ability to sleep during the morning, and sometimes in the afternoon as well. In contrast, those with chronic insomnia don't find it much easier to sleep during the morning than at night. Another important difference is that the DSP individual falls asleep at more or less the same time every night, and sleep comes quite rapidly if the person goes to bed near the time he or she is usually falls asleep. Young children with DSPS resist going to bed before they are sleepy, but the bedtime struggles disappear if they are allowed to stay up until the time they usually fall asleep.
Another fairly common disorder which causes sleep-onset insomnia is Restless Legs Syndrome (RLS). Individuals with RLS experience uncomfortable sensations in their lower legs at night, which make them want to move their legs or get out of bed and walk around.