When performing GAM and GLM fits to the same data set, I get an almost identical fit in terms of fitting metrics. However, the variables that are identified as significant differ between the two approaches. How do decide which method is most appropriate?
The reason that the significance of the coefficients differs is because, under the modeling assumptions, the calculation of standard errors differs. The correct answer is to choose the model that is most appropriate for--not the data--the scientific question at hand.
For instance, suppose I am interested in measuring a difference in rates of some disease in communities before and after applying some intervention. I could model the additive rates using a Poisson GLM with the identity link. This presumes that the distribution of rates follows a Poisson distribution. On the other hand, I could model these rates using a simple t-test. This makes no assumption about the distribution of rates and accounts for the possible underdispersion due to infections within a community not necessarily having a constant inter-arrival time. These two models will estimate the same rate difference, but give different standard errors.
Which, then, is correct? Well, the adequacy of the underpinning assumptions is what is called into question. Therefore, you should endeavor to describe and rationalize the model that is appropriate for the data--whether or not that model happens to be significant.