The House is the chamber with actual power, while the Senate was intended to act as a check on the House and the executive. That's why the 17th amendment made such a mess: they were supposed to represent state interests, and especially having a balanced budget is much more a state interest than a popular interest.

In particular, for any bill requiring spending, the Senate can only amend a bill that started in the House. Impeachment must start in the house and is then tried in the Senate. The Senate can't nominate someone to office, they can only confirm a nominee presented by the executive.