Both coordinates are the same. As you are in the Southern Hemisphere a False Northing (of +10000000m) is usually applied to eliminate the negatives.
The utm package applies the false Northing by itself. For Proj you need to specify it:
>>> p = Proj(proj='utm', ellps='WGS84', zone='34H'zone=34, south=True)
>>> p(lon,lat)
>>> (261762.55892627264, 6209508.925593799)