from my understanding a tenement is a housing structure where living conditions are less important than capacity. The rooms are only barely large enough to fit a single bed into and toilets are showers are generally shared by all the residents -- if there are really any amenities at all. Characteristically in the United States, the buildings are poorly maintained, filthy, uncomfortable, dangerous, and filled with residents who can barely afford to live anywhere at all.
These buildings were mostly built near the turn of the century to house immigrants and others who were flocking to the major cities, thus the emphasis was on cramming as many people in there as possible, regardless of their safety or comfort.
Now a tower block is called an apartments, apartment building, or tower. The rooms are self contained and are generally as comfortable as one can afford. Though how much one pays for a decent apartment depends on where you live and who you know, as well as what you consider "decent." In urban areas apartments most often multi-story buildings, though in less dense areas then tend to be a enclosed "complex" of many two to three story buildings with swimming pools, laundries, recreation areas, and combined covered parking.