Of course bike shorts have some extra padding for the ride, but that padding takes extra time to dry out after the swim. By the time you hit the run you'll be contending with diaper rash.

Here are some pros and cons - see for yourself.
Bike Shorts: Pros
- More comfort over longer bike legs
- Good grip for non-slip on legs
- Extra padding
- Specifically designed for biking
Bike Shorts: Cons
- Extra padding collects extra moisture
- Not designed for running and swimming
- Extra padding can cause extra chaffing even when dry
- Designed with all three sports in mind
- Less chaffing on runs
- Reduced padding allows for easier running and swimming
- Less fluid retention in pad
Triathlon Shorts: Cons
- Less padding and comfort for longer bike rides
- Thinner materials may result in lesser product life. But remember, these are triathlon race shorts. You will only wear them a couple of times a year.
The Bottom Line
You'll want to do a few training rides in tri shorts (just to get used to it), however, 95% the time you should train in bike shorts and race in tri shorts.
You can endure triathlons in bike shorts, but the more competitive you get, triathlon-specific shorts are the way to go. Especially on longer distances. You may sacrifice some comfort on the bike, but it will pay for itself over the run when you aren't chafing your way to the finish line.