The League Cup is only for teams in the Premier League and the next three professional divisions of English football (a.k.a. ‘The Football League’). That’s 92 teams in total.
The FA Cup, on the other hand, is open to all teams in the Premier League, Football League and then levels one to five (or six?) of the national league system, which includes hundreds of amateur and part-time clubs from regional leagues and conferences. This year, 736 teams entered the competition.
Those clubs further down the chain start in the FA Cup at a much earlier date, whereas the PL sides only join in the third round. A big tie against one of these bigger clubs is a much sought after reward for the ‘minnows’ from outside the professional divisions. If they progress far enough, they might get a lucky draw against a big team like Chelsea or Southampton (hehe) and try their hand at ‘giant killing’. Of course, the League Cup also sees small clubs against big clubs, but no club is from lower than League Two, so it’s not quite as romantic.
Edit: You might also be interested to know about the Football League Trophy (currently known as the JPT), which is open only to the teams in League One and League Two. There is also the FA Trophy (open to all teams in Levels 1 – 4 of the national league system) and the FA Vase (open to all teams in levels 5 – 8).