The t-test simply a special case of the F-test where only two groups are being compared. The result of either will be exactly the same in terms of the p-value and there is a simple relationship between the F and t statistics as well. F = t^2. The two tests are algebraically equivalent and their assumptions are the same.
In fact, these equivalences extend to the whole class of ANOVAs, t-tests, and linear regression models. The t-test is a special case of ANOVA. ANOVA is a special case of regression. All of these procedures are subsumed under the General Linear Model and share the same assumptions.
- Independence of observations.
- Normality of residuals = normality in each group in the special case.
- Equal of variances of residuals = equal variances across groups in the special case.
You might think of it as normality in the data, but you are checking for normality in each group--which is actually the same as checking for normality in the residuals when the only predictor in the model is an indicator of group. Likewise with equal variances.
Just as an aside, R does not have seperate routines for ANOVA. The anova functions in R are just wrappers to the lm() function--the same thing that is used to fit linear regression models--packaged a little differently to provide what is typically found in an ANOVA summary rather than a regression summary.