Why is there a difference? Because they mean two different things!
If something varies directly, then when one side of the equation increases, the other side increases as well.
If X varies directly with Y, then when Y increases, so does X
Let's say you have a lemonade stand. Each pitcher of lemonade can fill 8 glasses:
Pitcher = 8 Glasses
If you had 2 pitchers, you'd be able to fill 16 glasses.
Thus, the glasses of lemonade that can be sold is directly proportional (or varies directly) with the number of pitchers of lemonade.
On the other hand, inverse variation means the opposite. When two things are inversely related, as one side of the equation gets bigger, the other side of the equation gets smaller.
If X and Y vary inversely, then as X goes up, Y goes down, or as X goes down, Y goes up.
Using the same lemonade analogy:
The more glasses of lemonade you drink, the fuller you get and the less thirsty you are.
Glasses of Lemonade Drank = 1/Thirst
In other words, as you drink more glasses of lemonade, the left hand side of the equation gets bigger. In order to make the right hand side of the equation equally big, the denominator must get SMALLER. So as you drink more lemonade, you get less thirsty and the fraction gets bigger
On the other hand, if you drink less lemonade (left hand side of the equation gets smaller), then you'll get thirstier... this causes the denominator of the fraction on the right side to get bigger and thus the overall fraction gets smaller
Maybe that wasn't the best analogy. The direct proportion is easy to see, but I wanted to keep it in context, so my apologies if the inverse relationship seems a bit difficult!