Or, in other words, what kind of data can be gathered from inside tornadoes and how could it help in predicting them? Also, why isn't it possible to predict/estimate this kind of data from readings taken from outside the funnel (or is that what is currently done)?
Generally speaking, the types of data gathered outside a tornado can also be measured inside the tornado. Currently, forecasters try to estimate an area where tornadoes usually are. The forecasting time for tornadoes, or warn time, is about 13 minutes. There is still quite a bit we don't know about tornadoes, such as why some storms that look like they can produce a tornado, will not produce a tornado.
Currently, most data taken about tornadoes is taken outside of the tornado. Data from inside the tornado, while uncommon, can give improved estimates of wind speeds, and pressure changes, with a level of detail that is hard to get from the outside, looking in. The models we deal with are imperfect- there are too many sources of error and It generally doesn't help the current predictions, but research on that data may help future predictions.