Betting everything maximizes your mean wealth. Using the Kelly Criterion maximizes your median wealth.
Let's say there are 1024 (2^10) people with $1 each, and a bet which is 50 / 50 of returning either 3x or 0x (i.e. you bet $1 and either get back $3 or $0). Let's further specify that there will be 10 rounds of betting.
Using the "bet everything" strategy, you end up with the following distribution of outcomes:
10 wins, 0 losses -- 1 person: $59,049
1 or more losses -- 1023 people: $0
The average ending balance is $57.67, while the median ending balance is $0.
Using the Kelly Criterion, one will bet 25% of the bankroll each time. You will end up with the following distribution of outcomes.
0 wins, 10 losses -- 1 person: $57.66
1 wins, 9 losses -- 10 people: $28.83
2 wins, 8 losses -- 45 people: $14.41
3 wins, 7 losses -- 120 people: $7.20
4 wins, 6 losses -- 210 people: $3.60
5 wins, 5 losses -- 252 people: $1.80
6 wins, 4 losses -- 210 people: $0.90
7 wins, 3 losses -- 120 people: $0.45
8 wins, 2 losses -- 45 people: $0.22
9 wins, 1 losses -- 10 people: $0.11
10 wins, 0 losses -- 1 person: $0.05
The average ending balance is $3.25, while the median ending balance is $1.80.
Incidentally, the choice of maximizing the median is somewhat arbitrary. You can also use a similar approach to maximize, say, the 25th percentile outcome, at the expense of average and median outcomes (You would bet 10% of the stake each time, yielding a 25th percentile of $1.08, a median of $1.25, and a mean return of $1.28).