The reason why my distance is off, is because I only compared the first feature. This will code will give the same results as in QGIS:
import ogr
driver = ogr.GetDriverByName('ESRI Shapefile')
lineshp = driver.Open('U:/My Documents/Tool/shp/line.shp', 0)
linelyr = lineshp.GetLayer()
pointshp = driver.Open('U:/My Documents/Tool/shp/point.shp', 0)
pointlyr = pointshp.GetLayer()
linefeat = linelyr.GetNextFeature()
pointfeat = pointlyr.GetNextFeature()
point_geom = pointfeat.GetGeometryRef()
distlist = []
while linefeat:
line_geom = linefeat.GetGeometryRef()
dist = point_geom.Distance(line_geom)
distlist.append(dist)
linefeat.Destroy()
linefeat = linelyr.GetNextFeature()
print min(distlist)