Live Plant Pros
The fish seem to enjoy them more and will nibble at them.
The plants will use nutrients that the fish have expelled.
Plants grow and multiply so your not having to purchase more everytime you get a new tank.
Provide shade and shelter for the inhabitants and fry have a better chance of surviving predation.
Fish look great darting in and out of the real foliage.
Can make some pocket money selling on extra plants.
Live Plant Cons
If you get the wrong plants you could be destined for failure, too many places sell the unsuspecting nonaquatic plants.
Plants grow and multiply so maintenance is needed especially for quick growers.
Some plants will need fertilisers and Co2, which can reek havoc with other water paramaters if not done correctly.
Can harbour unwanted hitch hikers.
Fake Plant Pros
Always look good
No special care needed
Fish that normally demolish live plants wont destroy them.
Fake Plant Cons
Look good until algea grows all over them and then need cleaning
Can injure fish if using some of the hard plastic types of fake plants.
Never really look natural.
COULD (hence COULD) release petrochemical chemicals into the water long term if the plants where made cheaply.
Personally I use live plants in my tanks and I don't run Co2 or add fertilisers.