What's the difference between Vanguard Marks and Vanguard Reputation?

Solution 1:

Vanguard Reputation is just a barrier to allow you to purchase the Purple Gear from your class vendor. The Purple Gear requires you to be at least Level 2 Reputation, so farm those Bounties.

You use Vanguard Marks to actually buy the items - so farm those Strikes!

Think of Vanguard Reputation as your credit score, you need it high enough to qualify for a mortgage. You spend your Vanguard Marks on the downpayment. :)

Solution 2:

It's basically the difference between experience points and money.

Vanguard Reputation is "experience" used to "level up" your Vanguard vendors. You can buy better gear from them on higher levels, and you might need level 1 or 2 to be able to buy interesting gear in the first place.

Vanguard Marks is "money" used to purchase items from the Vanguard vendors.

Same difference exists between Crucible Reputation and Crucible Marks.