"Opening the house" refers in the first case to having a house prepared for habitation. When a family had both an estate in the country and a house in town, it was customary for one to be kept "closed" - sheets over the furniture to prevent them from getting dusty, most rooms unused, just a few servants kept on at half-pay in order to make the house somewhat inhabited and less likely to be burgled - when the family was living in the other one. In this particular case, if Robert had wanted to stay at his townhouse then Cora would have called the servants there and told them to remove the dust covers, prepare the larder and kitchen to make fancier food, and make sure that his bedroom was nice and cozy. That would be a lot to do for one person just staying a night or two, and it was simpler for him to stay at his club.

The London townhouse/country estate combo was a long-standing tradition of the English aristocracy; it began in the sixteenth century, when Henry VIII took over a number of bishops' palaces in the London area and gave or sold them to some of his most important courtiers. A generation or two later, courtiers would be building their own more private and modern residences - smaller in comparison to the episcopal palaces, but, I mean, we're talking about structures like Burlington House and Montague House. Those who couldn't afford to build places like this would typically rent substantial homes instead. All of them wanted to be in London for the cooler months, which became a more interesting time to be in town through a vicious cycle: because the aristocracy was there from autumn to spring, avoiding the heat of the summer in the country, there was more going on socially during the period, so more people wanted to be in London for "the season", so theaters and other venues put out more entertainments in the winter ... Anyway, this is the context for having a second home in London and needing to open it.

When the Crawleys hold an open house at Downton, that's a different matter. As is discussed in the show, the state of the English country house in the early twentieth century, particularly following the first World War, was fairly bleak. Many were sold to wealthy businessmen like Sir Richard Carlisle as impressive showpieces, which was a good fate: between 1920 and 1939, 221 mansions were demolished for various reasons (to avoid the expense of renovating, to allow the land to be subdivided, etc.), in comparison to 79 between 1870 and 1919. A good number of those that survive became museums under the National Trust, or were put on the National Heritage List to prevent much alteration, although I believe the bulk of the historic-house listing took place following World War II. By literally opening the house up to visitors and charging admission, the Crawleys were attempting to establish it as an historic site that should be preserved as an attraction while retaining autonomy.

So, to answer your basic question: in the first case they were discussing having rooms put in order for habitation; in the second they were discussing throwing the doors open to any visitors who wanted to come by.