The answer to this would depend in large part on how many seats were held by the one or two small parties other than the Republicans and the Democrats. If those parties held enough seats, the workings of Congress might be somewhat different.
The basic framework of Congress might or might not remain as is. In our current system, the Speaker of the House, for example, is chosen by a vote of the members that always goes on strict party lines. If no party held a majority of the seats, this system might be disrupted. If all the parties simply voted for their own leaders, there would be no change as the party with the most votes would win. But it might happen differently.
Let's assume the Democrats (as they are now) were in the minority in the House. They might make a deal with one of the smaller parties in which the smaller party would vote for the Democratic leaders in exchange for certain concessions. At this point, we would have a coalition government in the House where the Democratic Party had to be attentive to the needs of the other party.
In practice, however, this might not be such a big change. Right now, there are factions within parties that have to coexist. The freshmen members of the House Republican Caucus are said to be very ideologically different than the leadership. This means that the House leadership is already having to be attentive to the needs of a bloc of members whose views are different from their own. When looked at in this way, the presence of smaller parties might not actually make any difference at all.
All this is to say that it is hard to know if there would be any really important differences if we were to move to a situation in which more parties were represented in Congress.