Are boosted decision trees the same as a boosted random forest?

short answer: No.

longer answer: Random forests use the average of multiple trees that are trained in a way to reduce the correlation between trees (bagging with modified trees). Boosting trains sequentially, with each classifier working on the resulting residuals so far.

I am assuming that you meant boosted decision trees, sometimes gradient boosted decisions trees, as usually one have boosted decision trees. I think xgboost added boosted RF, and you can boost any supervised model, but it is not usual.

The training process differs, but the resulting model only differs in data rather than code -- you evaluate a bunch of trees and add them up.

For better or for worse (usually for better), boosted decision trees work harder to optimize the tree structure for a given problem. Random forests rely on enough trees being good enough.

Ignoring tree split selection, one technique people sometimes do makes the two techniques more related -- in gradient boosting, once the splits are chosen it's a sparse linear algebra problem to optimize the weights/leaves (iterative if your error is not MSE). That step would unify some part of the training between the two model types.