Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.
1. Let \( S \) be the number of students who study Spanish, \( F \) be the number of students who study French, and \( B \) be the number of students who study both languages.
2. We know the total number of students is 2001.
3. The number of students who study Spanish is between 80% and 85% of the school population:
   \[
   0.80 \times 2001 \leq S \leq 0.85 \times 2001
   \]
   Calculating these values:
   \[
   0.80 \times 2001 = 1600.8 \quad \text{and} \quad 0.85 \times 2001 = 1700.85
   \]
   Since the number of students must be an integer:
   \[
   1601 \leq S \leq 1700
   \]
4. The number of students who study French is between 30% and 40% of the school population:
   \[
   0.30 \times 2001 \leq F \leq 0.40 \times 2001
   \]
   Calculating these values:
   \[
   0.30 \times 2001 = 600.3 \quad \text{and} \quad 0.40 \times 2001 = 800.4
   \]
   Since the number of students must be an integer:
   \[
   601 \leq F \leq 800
   \]
5. Using the principle of inclusion-exclusion for the total number of students:
   \[
   S + F - B = 2001
   \]
   Rearranging for \( B \):
   \[
   B = S + F - 2001
   \]
6. To find the smallest number of students who study both languages (\( m \)), we need to minimize \( B \):
   \[
   B_{\text{min}} = \min(S + F - 2001)
   \]
   Using the smallest values for \( S \) and \( F \):
   \[
   B_{\text{min}} = 1601 + 601 - 2001 = 201
   \]
7. To find the largest number of students who study both languages (\( M \)), we need to maximize \( B \):
   \[
   B_{\text{max}} = \max(S + F - 2001)
   \]
   Using the largest values for \( S \) and \( F \):
   \[
   B_{\text{max}} = 1700 + 800 - 2001 = 499
   \]
8. Finally, we calculate \( M - m \):
   \[
   M - m = 499 - 201 = 298
   \]

The final answer is \(\boxed{298}\).