My impression:
Request First
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Pros: Don't have to give up the unit you already have until you find something better.
Cons: The further in advance you deposit your unit, the more trading power you have. As a result, your trade is continually losing power. (There is debate about how quickly this happens.)
When to use: You are willing to use the week you have but would consider "better" options. You want to shop more than one exchange service.
Example: You have a summer week in the mountains that is always fun, but if you could get a week in Maui, you would make the change.
Deposit First
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Pros: Fixes the trading power at the time of the deposit. Allows you to postpone the decision on when and where until well after the date of the week being deposited.
Cons: You give up control of your week and are somewhat at the mercy if II.
When to use: You don't want to use the week you have and feel that you can come up with a better choice through II.
Example: You have a week 50 in the mountains, but the kids are in school, snows not that good yet, and you're going later in the ski season anyway. Deposit it, and set up an ongoing search for Maui. If nothing comes up in the next year, trade for Orlando instead. The kids always enjoy that trip.