What are some specific responsibilities of the House members. I already know that the House acts as a "grand jury" in the case of impeachment of executive officials, and that all revenue bills must originate in the House. What else??
Unlike members of the Senate, members of the House are beholden to smaller constituencies. They are also backed by special interest groups as well as political parties, and there is a high incumbency rate. So, in addition to everything others have mentioned, it is important to note that there are over 400 senators in the House and each one has an agenda.
While the Constitution doesn't specifically state that spending bills must originate in the House, it has repeatedly tried to assert this authority over the course of our history.
In fact, when the Senate has originated a spending bill, passed it and sent it to the House, often times the House will simply attach a blue piece of paper to it arguing that the House's Constitutional prerogatives are being infringed upon. This practice is known as "blue-slipping".
Also all bills designed to raise revenue must originate from the House of Representatives and cannot start in the Senate. however the Senate can and does add on to the revenue bills once they reach the Senate.
Yep - the House plays an important role in case of no candidate receiving a majority. It is the House who decides the most important question of who will be president. Also, be aware of the point made in #3 - the House is supposedly more representative of the people, though this has been debated.
Well, perhaps it's too obvious to say, but it's the Speaker of the House who is third in line to the Presidency, following only the Vice President. This is undoubtedly so because the House is designed to be more representative of the people.
The House did not have nearly as many of these specific duties given to it as the Senate did since the House was not supposed to be as important.
The only one I can think of that you did not mention has to do with presidential elections. If no candidate gets a majority in the Electoral College, the House decides who will be president. Each state's delegation gets one vote.
The House not only decides who will be President in case of election problems, but the House is the legislative body where all bills originate and contain the most acts of responsibility as defined by the US Constitution. The House is supposed to be the body of the people represented by how many are in a district.