Where I teach, at the college level, I have noticed an increase in the use of worksheets, something that, in my opinion, is not necessarily a good trend, although they do have utility in some situations. The advantages and disadvantages of using worksheets depend on the content area and how they are being used, I think, in other words, a question of context.
First, how useful worksheets are is a matter of the course being taught. In subjects such as math, physics, or chemistry, they offer the student a means of practicing problem-solving, either in the classroom or as homework. In courses such as these, the students must have the opportunity to grapple with the problems presented, and this is a decided advantage. However, in subjects such as English, history, or sociology, most of the worksheets I have seen tend to focus on specific details, rather than encouraging students to see the bigger picture or to think critically. For example, if a worksheet includes questions on what character made what statement on what page, for an English course, I see that as a worksheet that is worse than useless, since it makes students believe that is what is important in studying literature. So, while worksheets provide an advantage in some content areas, they provide a significant disadvantage in others.
Second, the efficacy of worksheets is to a large degree a function of how the teacher uses them. If students spend all of their class time completing worksheets individually and the teacher sits at the desk, the disadvantages are many. Students will often lose their focus in a very short time, and they are meant to grapple, not to struggle, with what they do not know or understand. If this is what an occasional class consists of, with the teacher walking around offering individual student guidance, it could be fine. But that is often not the case. A better use of worksheets could be having students solving problems in groups, with the teacher walking around checking on progress and offering guidance. Worksheets can also be gone over with the class as a whole, so that a few representative problems are solved as a class. I do this when I teach accounting. If worksheets are handed out as homework, handed in the next class, and never gone over at all, this does not help students all that much. If they are making errors, they have no way of knowing until it's far too late for them. When worksheets are given as homework, at least a few representative problems should be covered with the entire class, to get everyone on the right track. I have also noticed that when teachers hand out worksheets, students infer that all that will be on exams will be the material reflected in the worksheets, and they tend to be quite offended when other material is on an exam. The worksheets have a decided negative effect upon student motivation to cover all the material in a course, focusing them on what they believe will be on a test. A teacher who intends to cover only worksheet material on exams is being true to the implied "contract," but doing his or her students a great disservice by limiting the areas of study and learning. It is far better to hand out a worksheet and tell one's students that the worksheet is meant to help them practice some of the material but that they are responsible for all material covered.
This over-reliance on worksheets does not mean that they are an evil in and of themselves. But we should always be aware of the context in which they are being used and our intentions in using them.