It is not exactly the same: even if all bounded subsets are totally bounded, there might be bounded sequence contains a Cauchy subsequence. But if you assume that the space is complete, then it is the same.
Proper space is the a complete space such that any bounded subset is totally bounded,
- or equivalently, in which any bounded sequence contains a Cauchy subsequence,
- or equivalently, any bounded closed set is compace,
- or equivalently, the distance function from one (and therefore any) point is proper; the latter means that invese image of any compact set is compact.