i flew ryanair (FRA-Hahn to Glasgow) & it was as organized as one would expect. bad thing though, i didn't get to sit next to me traveling partner cause they don't reserve seats. it's a free-for-all to get in the plane.
The embarrassing scramble for identical seats is probably the biggest thing that puts me off flying with budget airlines. That's not the fault of the airlines (there's no particular reason why one needs a specific seat reserved); it's just the sheep mentality that messes up the plan.
The other disadvantages of budget airlines, beyond those already mentioned here, are fewer flights to choose from, generally less convenient times and a lack of backup for when things go wrong. If a plane is faulty or can't land because of bad weather, budget airlines have fewer possibilities to get you to your destination than the traditional airlines.
Advantages include the ability to buy singles (this is revolutionary and fantastic), the fact that the provincial airports are sometimes more convenient (Frankfurt Hahn for the Eiffel and Friedrichshafen for Bodensee, for example), plus of course the lower prices. But the latter comes with a big caveat: traditional airlines have responded to the low-cost revolution by slashing their prices, and now often work out much better value. Always compare prices; don't assume that a budget airline is the cheapest.
You asked about both Ryanair and Easyjet, but these two airlines are very different. Easyjet, for example, usually do fly to a city's main airport; the Stansted routes in London are a relic of BA's former budget airline, Go!, which Easyjet bought up some years ago. Ryanair, on the other hand, always use out-of-the-way airports. There are now countless dozens of budget airlines so it's difficult to generalise, especially as the traditional airlines are making their short-haul flights more and more budget-like.