I'd say that statistics is a vast and open field and econometrics is... well economy. In fact econometrics is based in statistical models applied to financial and economical situations. These models include mostly time series models, vector autoregressive models and smoothing. Certainly it is quite interesting. In stats you study something beyond those models, you also work with surveys, multivariate methods, other models typicals for clinical studies or social sciences such as mixed models, hierarchical models, regression, etc.. In my humble opinion, econometrics is just a bit statistics with tons of economic meaning, because the field requires some deeper understanding of these subjects. It is quite cool indeed since it is a somehow "new" at least compared to statistics. In fact, it wasn't until the 60's, thanks to Samuelson, that math started to be an important part of finance and economics.