sp_updatestats just checks if the modification counter has a value greater than zero - it does not look at how many rows have been updated. It also uses a sampling rate by default - unless you utilize the 'RESAMPLE' option which will use the previous sampling rate that was used.
Auto update stats also uses a sampling rate - but it only kicks in when a certain threshold has been reached. This sampling rate may or may not be good enough depending on the distribution of data in the column.
My recommendation would be to build your own code - and determine what thresholds are valid and whether or not a sampling rate for that table is valid for your system. That is the only way to be sure that you are updating the stats as needed for your application.