Bagging - Bagging has a single parameter, which is the number of trees. All trees are fully grown a binary tree (unpruned) and at each node in the tree one searches over all features to find the feature that best splits the data at that node.
Random Forest -
Random forests has 2 parameters:
1-The first parameter is the same as bagging (the number of trees)
2-The second parameter (unique to random forests) is mtry which is how many features to search over to find the best feature. this parameter is usually 1/3*D for regression and sqrt(D) for classification. thus during tree creation randomly a mtry number of features are chosen from all available features and the best feature that splits the data is chosen.
Hope this helps!