Lotteries define a pick-3 straight bet as putting all three digits in exact order and define a boxed bet as the same three digits being in any order. When we box three digits for a $1, we are actually making 6 straight bets at a reduced cost; $1 divided by 6 equals $.167 and the payoff is $83. Some states round off the $.167 to $.16 and the payoff is $80. If you're using one digit twice in a box, you're making 3 straight bets at the reduced cost and the payoffs are $166, $167, or $160 on a $1 bet.
Wheeling and boxing the same digits accomplish the same thing but as Surge and Jani pointed out, wheel bets can only be made as three or six straight bets in amounts of $.50 or $1.00. If you ask the clerk for $1 box on 123 it will cost $1 and a $1 wheel bet will cost $6. Because the lotteries round off the box payoffs lower, a $6 box only pays $498 ($83 times 6) or $480 ($80 times 6) but for the same $6 you can make a wheel bet and the payoff is $500.
Boxing or playing three digits straight depends on how players choose their digits. Some players choose their best four or five digits and play 4 or 10 boxes with all those combinations and others might decide one digit might hit only one of the three positions and wheel it with the other chosen digits. For instance they could wheel the digit 1 in the first position with the digits 2, 3, and 4 in six straight combinations (123, 124, 132, 134, 142, 143) or use four digits with the digit 1 in the first position in twelve straight combinations.
There is no different in a straight payoff whether the combination has three different digits, two digits the same, or all three digits the same. A box bet is just another betting option that players can use with straight bets or use separate.
The idea of wheeling is to put the digits in the exact order and get the full straight payoff. In pick-4 games where the straight payoffs are higher, "boxed wheeling" can be used. If I play the boxed combination 1234, I could box the digits 1 and 2 in the first two positions with the digits 3 and 4 in the last two position and create a wheel with four straight combinations (1234, 1243, 2134, 2143). If I were just using four digits, I would spend the extra buck or 50 cents on the box bet, but I could box three digits in first two positions with two or three digits in the last two positions and create a wheel with 24 or 36 straight combinations. Most pick-4 games offer 50 cent wagers and betting $12 or $18 to win $2500 might seem more logical than betting the same amount to win at best $400 by playing multiple boxes.
It really comes down to player preferences on the amount to wager compared to the possible win expectation.