Generally the Senate is viewed as the more influential and prestigious body. A primary reason for this is its greater exclusive powers. In particular, the Senate has the sole power to ratify treaties, with a 2/3 majority required. For example this power was used in 2010 to ratify the New Start Treaty, passing in a 71-26 majority. In essence, the Senates role in foreign policy is far more extensive. Another of the Senate’s exclusive powers is to ratify the president’s nominations and this manifests perhaps most importantly regarding the Supeme Court. Not only can they vote against a nomination e.g when Robert Bork, the conservative justice nomination by Reagan in 1987 was voted against by 58-42 but also the mere threat of a veto severely affects who the president will even consider putting forward. For example Obama could now theoretically put forward a very liberal Supreme Court justice however he knows it would eb a wasten of time given the Republican majority in the Senate. However it should be remembered that the House too has its own exclusive powers such as in itiating money bills and the power of impeachement.
Moreover, in terms of individual factors the Senate clearly appears more presitigous and influential. Firstly, there are just 100 senators compared to 435 members of the House which automatically gives indivisuals a greater infleucne. Equally, Senate terms are 6 years compared to 2 in the House, so Senators can focus on their actual job and getting things done rather than just preparing for another coming election cycle. Lastly, as a political platform the Senate is generally viewed as higher, with representatives often looking to becoming Senators but never the other way around. Combine that with the fact that the Senate is generally the springboard for presidents e.g Obama, Nixon and Johnson all came from here, it is clear this chamber wields greater significance.
On the other hand, it could be argued that they are indeed equal in power because for their primary function i.e legislating, they have equal powers. In other words, they both require a 2/3 majority to pass a bill, they both require a 2/3 majority to iniatie amendments on bills and they both require a 2/3 majority to override presidential vetoes e.g in 2008 when Bush tried to veto legislation protecting doctors from a 10.6 percent cut in their reimbursement rates when treating Medicare patients. In short, a bill generally has as much or little chance of being accepted/rejected/amendments in either one of the chambers.