What separates easy global problems from hard global problems on graphs of bounded treewidth?
Most algorithms for graphs of bounded treewidth are based on some form of dynamic programming. For these algorithms to be efficient, we need to bound the number of states in the dynamic programming table: if you want a polynomial-time algorithm, then you need a polynomial number of states (e.g., n^tw), if you want to show that the problem is FPT, you usually want to show that the number of states is some function of treewidth. The number of states typically corresponds to the number of different types of partial solutions when breaking the graph at some small separator. Thus a problem is easy on bounded-treewidth graphs usually because partial solutions interacting with the outside world via a bounded number of vertices have only a bounded number of types. For example, in the independent set problem the type of a partial solution depends only on which boundary vertices are selected. In the Hamiltonian cycle problem, the type of a partial solution is described by how the subpaths of the partial solution match the vertices of the boundary to each other. 
Variants of Courcelle's Theorem give sufficient conditions for a problem to have the property that partial solutions have only a bounded number of types. 

If a problem is hard on bounded-treewidth graphs, then it is usually because of one of the following three reasons. 


There are interactions in the problem not captured by the graph. For example, Steiner Forest is NP-hard on graphs of treewidth 3, intuitively because the source-destination pairs create interactions between nonadjacent vertices.


Elisabeth Gassner: The Steiner Forest Problem revisited. J. Discrete Algorithms 8(2): 154-163 (2010)

MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, Dániel Marx: Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth. J. ACM 58(5): 21 (2011)


The problem is defined on the edges of the graph. Then even if a part of the graph is attached to the rest of the graph via a bounded number of vertices, there could be many edges incident to those few vertices and then the state of a partial solution can be described only by describing the state of all these edges. This is what made the problems in [3,4] hard.
Each vertex can have a large number of different states. For example, Capacitated Vertex Cover is W[1]-hard parameterized by treewidth, intuitively because the description of a partial solution involves not only stating which vertices of the separator were selected, but also stating how many times each selected vertex of the separator was used to cover edges.


Michael Dom, Daniel Lokshtanov, Saket Saurabh, Yngve Villanger: Capacitated Domination and Covering: A Parameterized Perspective. IWPEC 2008: 78-90