"Ordinary linear regression" has perhaps a thousand different formulations. In the optimization world, it is called least squares since it minimizes the squared error of the residuals. It also maximizes the likelihood of bivariate normal data. Intuitively, this gives a pretty good "line of best fit" on a XY scatter plot of your data.
In the generalized linear model world, a OLS model is called a GLM with Gaussian error structure and identity link. GLMs use probability models for the outcome to formulate a maximum likelihood based approach to estimation. There are other GLMs to maximize likelihood for binomially distributed outcomes, Poisson distributed outcomes, etc. These do not minimize the squared error since they take account of a mean-variance relationship.
In short, GLMs are a family of regression models that depend on the type of probability model for the outcome, inclusive of ordinary linear regression. If the outcome is a count or a proportion, you should consider a GLM specially designed to that purpose.
The choice of whether to use a GLM depends on the nature of the data, not the results obtained from a model. In my work, when I fit a bad model, I consider the data and whether I "missed" something, but most often I resolve to simply report the "bad" results, such as "We did not find evidence of association between X and Y, a scatter plot revealed no clear trend between the values, and the r^2 for the fit was very low (r^22=0.15). A larger sample may reveal a more significant trend, but confidence intervals for the regression slope indicate that this would be at most X.X-X.X which is not meaningful.".