There are some terminology differences where the same thing is called different names in different disciplines:
- Longitudinal data in biostatistics are repeated observations of the same individuals = panel data in econometrics.
- The model for a binary dependent variable in which the probability of 1 is modeled as $1/(1+\exp[-x'\beta])$ is called a logit model in econometrics, and logistic model in biostatistics. Biostatisticians tend to work with logistic regression in terms of odd ratios, as their $x$s are often binary, so the odd ratios represent the relative frequencies of the outcome of interest in the two groups in the population. This is such a prevalent interpretation that you will often see a continuous variable transformed into two categories (low vs. high blood pressure) to make this interpretation easier.
- Statisticians' "estimating equations" are econometricians' "moment conditions". Statisicians' $M$-estimates are econometricians' extremum estimators.
There are terminology differences where the same term is used to mean different things in different disciplines:
- Fixed effects stand for the $x'\beta$ in the regression equation for ANOVA statisticians, and for a "within" estimator for econometricians.
- Robust inference means heteroskedasticity-corrected standard errors for economists (with extensions to clustered standard errors and/or autocorrelation-corrected standard errors) and methods robust to far outliers to statisticians.
- It seems that economists have a ridiculous idea that stratified samples are those in which probablities of selection vary between observations. These should be called unequal probability samples. Stratified samples are the ones in which the population is split into pre-defined groups according to characteristics known before sampling takes place.
- Econometricians' "data mining" (at least in the 1980s literature) used to mean multiple testing and pitfalls related to it that have been wonderfully explained in Harrell's book. Computer scientists' (and statisticians') data mining procedures are non-parametric methods of finding patterns in the data, also known as statistical learning.
I view the unique contributions of econometrics to be
- The ways to deal with endogeneity and poorly specified regression models, recognizing, as mpiktas has explained in another answer, that (i) the explanatory variables may themselves be random (and hence correlated with regression errors producing bias in parameter estimates), (ii) the models can suffer from omitted variables (which then become part of the error term), (iii) there may be unobserved heterogeneity of how economic agents react to the stimuli, thus complicating the standard regression models. Angrist & Pischke is a wonderful review of these issues, and statisticians will learn a lot about how to do regression analysis from it. At the very least, statisticians should learn and understand the instrumental variables regression.
- More generally, economists want to make as few assumptions as possible about their models, so as to make sure that their findings do not hinge on something as ridiculous as multivariate normality. That's why GMM is hugely popular with economists, and never caught up in statistics (even though it was described as the minimum $\chi^2$ by Fergusson in late 1960s). That's why adoption of empirical likelihood grew exponentially in econometrics, with but a marginal following in statistics. That's why economists run their regression with "robust" standard errors, and statisticians, with the default OLS $s^2 (X'X)^{-1}$ standard errors.
- There's been a lot of work in the time domain with regularly spaced processes -- that's how macroeconomic data are collected. The unique contributions include integrated and cointegrated process and autoregressive conditional heteroskedasticity ( (G)ARCH ) methods. Being generally a micro person, I am less familiar with these.
Overall, economists tend to look for strong interpretation of coefficients in their models. Statisticians would take a logistic model as a way to get to the probability of the positive outcome, often as a simple predictive device, and may also note the GLM interpretation with nice exponential family properties that it possesses, as well as connections with discriminant analysis. Economists would think about the utility interpretation of the logit model, and be concerned that only $\beta/\sigma$ is identified in this model, and that heteroskedasticity can throw it off. (Statisticians will be wondering what $\sigma$ are the economists talking about, of course.) Of course, a utility that is linear in its inputs is a very funny thing from perspective of Microeconomics 101, although some generalizations to semi-concave functions are probably done in Mas-Collel.
What economists generally tend to miss, but, IMHO, would benefit from, are aspects of multivariate analysis (including latent variable models as a way to deal with measurement errors and multiple proxies... statisticians are oblivious to these models, though, too), regression diagnostics (all these Cook's distances, Mallows $C_p$, DFBETA, etc.), analysis of missing data (Manski's partial identification is surely fancy, but the mainstream MCAR/MAR/NMAR breakdown and multiple imputation are more useful), and survey statistics. A lot of other contributions from the mainstream statistics has been entertained by econometrics and either adopted as a standard methodology, or passed by as a short term fashion: ARMA models of the 1960s are probably better known in econometrics than in statistics, as some graduate programs in statistics may fail to offer a time series course these days; shrinkage estimators/ridge regression of the 1970s have come and gone; the bootstrap of the 1980s is a knee-jerk reaction for any complicated situations, although economists need to be better aware of the limitations of the bootstrap; the empirical likelihood of the 1990s has seen more methodology development from theoretical econometricians than from theoretical statisticians; computational Bayesian methods of the 2000s are being entertained in econometrics, but my feeling is that are just too parametric, too heavily model-based, to be compatible with the robustness paradigm I mentioned earlier. Whether economists will find any use of the statistical learning/bioinformatics or spatio-temporal stuff that is extremely hot in modern statistics is an open call.
