I am pretty sure this topic has been discussed many times and is there a clear winner ... I guess not.
I have been working with waterfall methods all my life and more recently with agile, if I have to summarise they both have their advantages and disadvantages.
I thoroughly enjoyed doing analysis in a water fall method, partly because you were given the opportunity to think and discuss solution upfront, so more time was spend doing the requirements gathering this works really well where project has limited resource (time or money) and clear objective.
On the other hand the disadvantage with the waterfall method (more so in the current era) is that it is not as versatile as agile. Business are growing (and some of them are shrinking), as day goes by there are new competition in the market, new legislations and other factors which affect the company and needs to be responded immediately, in this type of environment agile allows quick and timely changes.
Agile also allows team to delivery solution to the QA team and possible to the customer quicker than the waterfall method.