Difference between Bash shell and Bash terminal?
There is no difference, they are both instances of the bash shell.
The reason you are seeing a different prompt is that your initial login shell sources ~/.bash_profile where presumably you have your prompt set. When you type bash it invokes another shell but because this one isn't a login shell, it doesn't source ~/.bash_profile and so you get the default prompt.
If you were call bash -l, (which invokes bash as if it were a login shell) I bet you would see that your original prompt remains