A generalized linear model specifying an identity link function and a normal family distribution is exactly equivalent to a (general) linear model. If you're getting noticeably different results from each, you're doing something wrong.
Note that specifying an identity link is not the same thing as specifying a normal distribution. The distribution and the link function are two different components of the generalized linear model, and each can be chosen independently of the other (although certain links work better with certain distributions, so most software packages specify the choice of links allowed for each distribution).
Some software packages may report noticeably different $p$-values when the residual degrees of freedom are small if it calculates these using the asymptotic normal and chi-square distributions for all generalized linear models. All software will report $p$-values based on Student's $t$- and Fisher's $F$-distributions for general linear models, as these are more accurate for small residual degrees of freedom as they do not rely on asymptotics. Student's $t$- and Fisher's $F$-distributions are strictly valid for the normal family only, although some other software for generalized linear models may also use these as approximations when fitting other families with a scale parameter that is estimated from the data.
It's hard to be definitive without knowing all the details of your model (such as sample size), but I would remark that the likelihood ratio, Wald, and score estimators are only asymptotically equivalent. That is, they agree as N $\rightarrow$ $\infty$.
The Wald estimator is generally considered to be the least reliable in terms of Type I and Type II error. In GLM applications, the likelihood ratio is to be preferred. Additionally, the Wald is not always consistent under transformations. Numerous studies abound that confirm these features, and I have even run simulations of my own using ecological data.
Regarding the limitations of the Wald test, see, for example:
Fears, Thomas R.; Benichou, Jacques; and Gail, Mitchell H. (1996); A reminder of the fallibility of the Wald statistic, The American Statistician 50:226–7
However, there are situations in which the Wald might behave better. I am not an expert on these statistical approaches, but here's an example:
Yanqing Yi and Xikui Wang (2011). Comparison of Wald, Score, and Likelihood Ratio
Tests for Response Adaptive Designs, Journal of Statistical Theory and Applications, 10(4): 553-569
Hope that helps,
Brenden
Best Answer
For a conditional normal distribution, the result would indeed be in line with the normal linear model.
Example in R
For all other distributions in the GLM family (e.g. Gamma, Poisson or Bernoulli), the results would differ, e.g. by taking into account the variance heterogeneity that is implied by the distributional family and also by different numerical techniques (iteratively reweighted least-squares instead of a single least-squares iteration).
So e.g. for the Gamma:
This is an additive model for a response with conditional Gamma distribution, correctly taking into account the non-homogeneity of the variance induced by the Gamma assumption.
While using an identity link with non-normal conditional response might lead to numerical instabilities in certain cases, it is a neat trick to e.g. adjust a difference in two proportions for confounders: to do so, you would run a logistic GLM with identity link.