Handicap System and League Manager have different functions, capabilities, and purposes.

Handicap System is intended for organizations that wish to "Handicap" a membership.  Most follow USGA rules and wish to calculate a certified USGA index to issue to their membership...which HS does...and follows exact USGA rules, including revisions of handicaps at a particular date, exceptional weight to tournament scores, etc. Of course Handicap System can also calculate a custom handicap where you can deviate from USGA and setup your own system. Scores shot on any course/tee can be quickly posted to any golfer at anytime.

League Manager also calculates a handicap...but it is in the "context" of running a league.  Most leagues have a set schedule of matches rotating across courses, set "event" dates, track standings (many use points to determine standings), some are also concerned about no-shows, substitutions, etc...and most also track a league handicap for their golfers.  So League Manager can also do handicapping.  In fact it can be setup to handicap any way you wish...including emulating the USGA method (best 10 of last 20 scores for handicap)  I say "emulate" because League Manager does not issue "official" certified indexes / handicaps.  League Manager does not get into the nuances of USGA rules for exception weight to tournament scores, revision date, etc....these are just beyond the scope of most leagues, and would add an unnecessary layer of complexity.  In fact the vast majority of  leagues do not use USGA as their method mainly because USGA uses 20 scores to base a handicap on.  For most leagues this does not allow a handicap to change fast enough. Plus USGA requires a minimum of 5 score before a handicap can be calculated, and most league want handicaps for new golfers established much sooner.

In conclusion, you could use League Manager to track handicaps for a group of golfers, but if you are really not running a league and you wish to just track handicap...Handicap System would be a better choice.

A good analogy would be you can use Microsoft Excel to write a letter....but it is really meant to work with spreadsheets...and Word would be a better choice.