The Speaker is a key role in the House, having a lot of control over the basic functions - without a Speaker nothing else can happen, and they control much of the flow of business once selected. Nancy Pelosi appeared to be very effective in this role, Kevin McCarthy has bargained away much of his own power to get elected - it remains to be seen how this works out.
Republicans control the House in the sense that they have the votes for a simple majority when they all work together, but as the Speaker situation has shown they are significantly divided. How well they exercise the control of the house is very much in question.
Without control of the Senate they are not going to be able to do anything significant but may pull off big symbolic moves only to have them blocked as they move through the rest of the system.
This could end up simply being a blockade to any further Democrat agendas until the next election which could swing things in either direction.
Hope that helps!
Sort of. Part of the problem is that in the Senate a simple majority is not usually enough - thanks to the very strange filibuster rule they usually need 2/3rds to pass something, and they needed two senators who seemed to enjoy obstructionism to reach that number. In that situation they were stuck, and so often made major concessions to their bills to appease the obstructionists.
The current situation is not conducive to big moves by either party but for symbolic action to raise outrage the Republicans are well placed. They often manage to turn outrage in to votes, so it may benefit them in the next election. For Democrats the opposite effect seems to happen more often from what I can see, so their position is less advantageous.