Mathematically there is no difference. As Adrian nicely pointed out: the ANOVA model is a special case of a regression model in which all the predictors are categorical.

But there is a difference in the application of "ANOVA" and "regression (analysis)" (deliberatily without "model") that has not been addressed in the answers above:

ANOVA is a tool to check how much the residual variance is reduced by predictors in (nested regression) models, whereas the regression analysis aims to quantify effect sizes in terms of "how much is the response expected to change when the predictor(s) change by a given amount?". For categorical predictors this reduces to the question to "what is the expected difference in the response between different groups/categories?". For continuous predictors this is the questions for a slope.

To clarify: ANOVA can be applied to any regression model (no matter if the model contains only continuous, only categorical, or both kinds of predictors). ANOVA allows to assess the impact of a predictor or a whole set of predictors on the residuals: who much of the variance in the data can be explained by these predictors? The regression analysis, on the other hand, is a complementary tool to asses the quantitative relation between a predictor and the response.

Generalization:

Regression models are (general) linear models (LM). (Multiple) linear regression is the analysis of a special case of a linear model (a model with one or several continuous predictor/s), The term "linear" in "linear model" refers to the estimated coefficients and *not* to the relationship between predictor and response. A linear model can model a curved relationship; only the estimated parameters of the model must be untransformed (i.e.: linear).*

Since "regression analysis" and "regression models" are often confused, I'd prefer to use the terrm "linear models". One may estimate parameters in such linear models and/or analyze the variance reduction due to (sets of) predictor(s) (-> ANOVA).

Further generalization:

For statistical models with non-normal errors, the term "variance" makes not much sense and is substituted by the more general "deviance" (http://en.wikipedia.org/wiki/Deviance_%28statistics%29). Models with error-distributions from the exponential family are then termed generalized  linear models (GLM), and the pendant to the ANOVA is the analysis of deviance. It gets even more universal when we consider generalized additive models (GAM) and verctorizes generalized or additive models (VGLM/VGAM).

* Hence, Y = A*sin(X) + B is a linear model for a non-linear relationship between X and Y, whereas Y = sin(A*X) + B is not a linear model (the parameter A is "inside" a function, so it is not linear anymore and the model can not be formulated to have both parametersin their linear form.

There is no difference, ANOVA is simply a special case of regression analysis where all of the predictor variables are categorical.

I partially agree with Adrian, ANOVA is a special case of regression, but from the perspective of their uses, there is a different flavour: if the independent/predictor variable is categorical, you must use ANOVA, otherwise use regression analysis.

On the other hand, when the dependent variable is dichotomous or categorical, you must use Logistic GLM.

Good question, and both Adrian and Jose give good answers! Let me just add that certain disciplines prefer the use of one over the other. For example, psychology has historically utilized ANOVA models, whereas economics has historically utilized regression. I personally find that regression is more flexible and intuitive, and rarely use ANOVA, except when comparing balance in baseline characteristics between multiple groups.

Mathematically there is no difference. As Adrian nicely pointed out: the ANOVA model is a special case of a regression model in which all the predictors are categorical.

But there is a difference in the application of "ANOVA" and "regression (analysis)" (deliberatily without "model") that has not been addressed in the answers above:

ANOVA is a tool to check how much the residual variance is reduced by predictors in (nested regression) models, whereas the regression analysis aims to quantify effect sizes in terms of "how much is the response expected to change when the predictor(s) change by a given amount?". For categorical predictors this reduces to the question to "what is the expected difference in the response between different groups/categories?". For continuous predictors this is the questions for a slope.

To clarify: ANOVA can be applied to any regression model (no matter if the model contains only continuous, only categorical, or both kinds of predictors). ANOVA allows to assess the impact of a predictor or a whole set of predictors on the residuals: who much of the variance in the data can be explained by these predictors? The regression analysis, on the other hand, is a complementary tool to asses the quantitative relation between a predictor and the response.

Generalization:

Regression models are (general) linear models (LM). (Multiple) linear regression is the analysis of a special case of a linear model (a model with one or several continuous predictor/s), The term "linear" in "linear model" refers to the estimated coefficients and *not* to the relationship between predictor and response. A linear model can model a curved relationship; only the estimated parameters of the model must be untransformed (i.e.: linear).*

Since "regression analysis" and "regression models" are often confused, I'd prefer to use the terrm "linear models". One may estimate parameters in such linear models and/or analyze the variance reduction due to (sets of) predictor(s) (-> ANOVA).

Further generalization:

For statistical models with non-normal errors, the term "variance" makes not much sense and is substituted by the more general "deviance" (http://en.wikipedia.org/wiki/Deviance_%28statistics%29). Models with error-distributions from the exponential family are then termed generalized  linear models (GLM), and the pendant to the ANOVA is the analysis of deviance. It gets even more universal when we consider generalized additive models (GAM) and verctorizes generalized or additive models (VGLM/VGAM).

* Hence, Y = A*sin(X) + B is a linear model for a non-linear relationship between X and Y, whereas Y = sin(A*X) + B is not a linear model (the parameter A is "inside" a function, so it is not linear anymore and the model can not be formulated to have both parametersin their linear form.

While I agree that ANOVA is a special case of regression, there are further differences in how ANOVA models tend to be applied. A classical ANOVA design has factors that are orthogonal and parameterised in a particular way (usually effect coding or similar coding scheme) with equal cell sizes (balance). As the factors are orthogonal, interaction terms are relatively easy to interpret and thus in a design with multiple factors is usual to estimate a full factorial model with all interaction terms. This approach would be problematic in a regression model with predictors that aren't likely to be orthogonal (and indeed causes arguments about how to partition variance in unbalanced ANOVA designs - where the same problems arise).

There is no major difference except for the use of it. ANOVA - traditionally speaking - is heavily used for experimental statistics, where there is a control group and an experiment group. You want to find out whether the variances and mean different from the population, statistically speaking. For regression, - traditionally speaking again - is heavily used in predictive modelling where accuracy of the model becomes a major concern.

ANOVA model is a special case of a regression model in which all the predictors are categorical

For clarity, It may be seen in the following reference.

Business statistics by examples fifth edition

by Terry Sinich

ANOVA is subset of Regression