A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides to use the promotion that will save her the most money. How many dollars does Jane save by picking one promotion over the other? (For example, if Jane spends $\$150$ on her purchase by using one promotion and $\$100$ on her purchase by using the other promotion, she saves $150-100=50$ dollars by using the second promotion over the first.)
Let's examine the two promotions individually.

Promotion A: Jane pays full price, or $\$30$ for the first pair of shoes. She pays a reduced price, $\$30\div 2=\$15$ for the second pair of shoes. Therefore, she pays a total of $\$30+\$15=\$45$ for the two pairs of shoes.

Promotion B:  Jane pays full price, or $\$30$ for the first pair of shoes. She pays $\$10$ off the full price or $\$30-\$10=\$20$ for the second pair of shoes. Therefore, she pays a total of $\$30+\$20=\$50$ for the two pairs of shoes.

Since Jane will pay $\$50$ if she chooses Promotion B and $\$45$ if she chooses Promotion A, Promotion A is the better deal. Jane saves $50-45=\boxed{5}$ dollars by picking Promotion A over Promotion B.