I just switched over, and what Cearbhaill says is true - I look at my plants and wonder why I ever liked plastic.
Benefits - plants use up excess nutrients in the water (as well as things like metals that are dangerous to fish), are absolutely beautiful to look at, fish can't cut themselves on a leaf like they can with plastic, they change and grow so your tank doesn't look the same day in and day out, and they offer a new dimension to the hobby - there is so much more to it than just fish!
Disadvantages - you'll need to cater to the needs of the plants as well as the fish - they need enough light, CO2, and fertilizer; some plants may not suit the conditions in your tank (with plastic you can choose any shape/colour you like, with real you have to buy based on suitable conditions, as with fish), you'll have to dedicate a bit of time to pruning them regularly if you get fast-growing plants (otherwise your fish will live in a jungle), plants can be initially expensive (though unlike plastic they grow, so soon that little tiny $12 plant may be 10" high and producing six plantlets).
Take a look online for information on planted tanks - there's a lot out there, and I think most people will tell you it's a little extra work for a lot of extra joy.