Here’s a really critical difference that no one has mentioned: If no candidate gets a majority of the electoral votes, the House gets to elect the President and the Senate the Vice President. Of course, this hasn’t happened in a presidential election since 1824 nor in a vice presidential election since 1836. But it’s interesting that in this one respect the House is more powerful.
It should be noted that the requirement that “revenue bills originate in the House of Representatives” is largely pro forma. All this really means is that revenue bills must come out of conference committee with an “H.R.” instead of an “S.” designation, which means the House gets to vote first on the conference report.
One thing. Minimum age for a Representative: 25. Minimum age for a Senator: 30. So you’d tend to see more politicians starting their careers running for the House. That, plus the numerical disparity between the chambers and the longer terms for Senators means that the House is more of a “warm-up bench” for the Senate. In other words, you’re less likely to see a Senator move to the House than see a Representative (with the attendant experience) move to the Senate.
I think the perception of the two bodies has changed with time. Henry Clay started his Congressional career in the Senate, filling a vacancy (and he wasn’t even 30) and then later he went to the House and even got elected Speaker as a freshman!
But in the early days of Congress, the House had a higher profile because the Senate used to conduct its business in secret. And since the senators were appointed, nobody had much of a connection with them. But when people like Clay and Webster and Calhoun moved over to the Senate, that body’s prestige increased.
Another problem with the House’s image is that there were no sitting House members who were elected President in the 20th Century. The last person to give up his House seat and get elected president was James Garfield and even he would have become a Senator if he hadn’t been nominated for President by the Republicans.