Why does it matter which committee in the House of Representatives a bill is sent to?

Because House Rule X demands it

The various standing committees shall have general oversight responsibilities as provided in paragraph (b) in order to assist the House in_ (1) its analysis, appraisal, and evaluation of_

(A) the application, administration, execution, and effectiveness of Federal laws; and

(B) conditions and circumstances that may indicate the necessity or desirability of enacting new or additional legislation; and

(2) its formulation, consideration, and enactment of changes in Federal laws, and of such additional legislation as may be necessary or appropriate.

Here are the applicable committees

Committee on Agriculture

Committee on Appropriations

Committee on Armed Services

Committee on the Budget

Committee on Education and the Workforce

Committee on Energy and Commerce

Committee on Financial Services

Committee on Government Reform

Committee on Homeland Security

Committee on House Administration

Committee on International Relations

Committee on the Judiciary

Committee on Resources

Committee on Rules

Committee on Science

Committee on Small Business

Committee on Standards of Official Conduct

Committee on Transportation and Infrastructure

Committee on Veterans' Affairs

Committee on Ways and Means

Permanent Select Committee on Intelligence