No, it doesn't matter much. The regulations for what can be done in a national monument or park (the only difference between the two is that one is established by presidential edict, the other by act of congress) are pretty strict, so there won't be huge differences in what you are allowed to do in one national monument compared to another.
That said, each different park or monument is run by different people who will invetably make different decisions about details. However, more significant differences arise due to the nature of the park/monument itself. Yosemite is relatively small, particularly the valley, and close to major population centers. As a result, it gets a lot of visitors, and this can't be ignored in formulating detailed policy. For example, the numbers of people in the valley are limited, and you need a permit to stay overnight in the back country.
Grand Staircase / Escalante, in contrast is large, farther from large populations, and the main attractions more spread out. There is no attempt to limit numbers, even in the most popular places like the Dry Gulch slot canyons. It's far enough and hard enough to get to that the numbers of visitors is way less than Yosemite Valley, so management policies can be much looser.
The general feeling is that the BLM is more laid back about things than the Park Service, but that is really hard to measure because of the kinds of lands the two agencies manage and what they are charged with. Would the Park Service manage Grand Staircase / Escalante all that differently? It's really hard to say since comparing it to places like Yosemite and Yellowstone is apples to oranges.
The BLM manages so few parks/monuments that it's hard to say whether there is a significantly different philosophy compared to the Park Service for lands that are mandated to have the same level of protection. My personal impression is that there isn't much difference. The few parks/monuments the BLM does manage are more out of the way and less major tourist attractions than many of the Park Service parks/monuments. To make any kind of fair comparison, you'd have to look at Park Service parks/monuments that are also similarly out of the way and less visited.
The BLM does manage a significant number of Wildernesses. These have even a higher level of protection than National Parks or Monuments. I've been to a number of them, and haven't really noticed much difference in how Wildernesses are managed by the BLM, the Forest Service, or the Park Service.
The one difference that does stand out is not in how the resource is managed or what you're allowed to do, but in interpretive programs. The Park Service has ranger-naturalists that lead guided hikes, give talks, and the like. This is either rare or non-existant in parks/monuments run by other agencies as far as I have experienced. Park Service parks/monuments are more likely to have a entrance fee, perhaps as a result of the extra services offered. Grand Staircase / Escalante does have a visitor center, which is actually quite nice and doesn't feel any different from visitor centers I've been to for lands run by the Park Service. However, I don't recall nature programs offered there as you would usually find in a Park Service park, although I wasn't explicitly looking for them when I was there last summer.