The Constitution differentiates the two houses in a couple of different ways.
All legislation that has to do with taxes or spending must originate in the House of Representatives. Once the bill has been proposed, both houses are on an equal footing. Since all bills enacted into law must pass both houses, there isn't any difference between them in terms of specialization. Both houses have their equivalent committees with certain Congressmen or Senators specializing in different areas and spending their times in committees. For instance, both houses have an Armed Services Committee, a Banking Committee, a Foreign Relations Committee, a Commerce Committee, an Intelligence Committee, and so on.
The Senate was given the power of advice and consent, meaning that all treaties negotiated by the president and all appointees made by the president must be approved by a majority vote of the Senate. That includes judges, cabinet members, ambassadors, and even senior officers in the military.
Impeachment is another area of differentiation. Any officer in the government can be impeached. With that process, the House votes on a majority basis to "impeach," i.e. indict. The House then appoints its members as the prosecution if impeachment happens. A trial then takes place in the Senate with the House members prosecuting the case. The Senate convicts on a 2/3 majority vote. Anything less is an acquittal. Two presidents have been impeached. Andrew Johnson (who followed Lincoln) avoided conviction by a single vote. Bill Clinton was impeached but was acquitted by the Senate.
If the electoral college fails to give a majority, the House votes on a state-by-state basis for president (each state gets one vote, so whoever has a majority in a state will vote for their candidate while a tie means a state gets no vote). The Senate votes for vice president on a majority vote.
Both houses are roughly equal in terms of power. The use of the terms "lower chamber" for the House and "upper chamber" for the Senate are just semantics and really have no meaning. Rules make the Senate much harder to pass legislation with the filibuster available for all but spending measures, while in the House, even a one-vote majority gives absolute iron-clad control to the party in power, so in effect, the House tends to be much more partisan while the Senate requires consensus on everything or nothing happens.
