Actually if drop measurement and leave only or narrowed question: What's the difference between method and procedure? A nice answer could be found here. From this answer we may conclude that method is a wider concept (more abstract) as opposed to a certain (more concrete) set of actions that the procedure is. You may think about the procedure as being a subset of method.
Measurement in this context is just a field where some particular examples live. For me it is easier to think of many examples in estimation methods, so any help form the community is welcome.
For example, in econometrics you often need to obtain data through the set of Monte Carlo simulations (that is a method). When I solve a particular simulation problem I first describe the data generating process, set the parameters of the experiment and run the simulations, that are than generalized into some averaged results. All this sequence is the procedure.
Most of the data collection methods are design based. So you have to work with one of the sampling methods. Take the first, simple random sampling (method) and apply it to a specific population sampling problem, the way you apply the method, a certain list of actions in this case will be a procedure of collecting your data.