I'm thinking that many people's perceptions of what constitutes word problems v. problem solving may be different than the perceptions of others. I've always described math problems that involve words as "story problems", even in calculus and physics classes. Many of those problems involved the cumulative knowledge of previous chapters' lessons along with the new lesson. These problems, at least in the math arena, always followed some increasingly difficult repitition in the functions being studied as well as those studied prior.
I'm kind of hoping that you could provide some kind of example of what you are describing when you refer to each. Not nit-picking; rather, I'm just trying to discern what you are thinking when you compare the two.
I will try to give an example of each. to me, a word problem or a story problem can usually be worked out in a minute, two at most. In fact, some children will quit if they can not get an immediate answer.
Word Problem
Sound travels through air at a speed of 1,129 feet per second. At this rate, how far will sound travel in 1 minute?
Simple arithmetic can solve this problem. Not much thinking is required - just do more of what that lesson has been about.
Problem Solving/Critical Thinking
Create a multiplication problem where the product is 45.89 and one of the factors is a whole number.
This second problem has more than one answer. It can create some discussion in the class. Students must think! Their reasoning and explanations can let the teacher see the level of understanding reached, or where there may be additional teaching needed.