Taking Prague as the most prominent example, apartments generally offer better value than hotels, in particular having additional living space and facilities.  For example even a studio apartment will have a private bathroom within the apartment, and at least a kitchenette.  A one bedroom apartment will generally have a kitchen area, a dining area, and a lounge/2nd sleeping room.  Larger apartments will have more bedrooms, a larger lounge, dining, and kitchen area, and sometimes additional bathrooms/wc's.   Ideal for families or small groups of friends.  The lounge will often have an additional bed and/or a sofa bed, to increase the sleeping capacity.

Prices are generally lower than hotels, and although hotels offer inclusive breakfast (which of course you pay for!) it costs very little to buy provisions to eat in the kitchen/dining area of your apartment, or you can have breakfast in one of the many cafes in Prague.  You may of course prefer the communal facilities that the larger, more expensive hotels can offer, but many of the smaller hotels in Prague, for example, are in old, renovated buildings which do not have that kind of space, and often just have a small dining room and bar at the most. 

Many of the best apartments are actually in the most central area, where the hotels are at their most expensive.

In other cities, like Budapest, apartments retain the same advantages, but in Paris we find that B & B's (Bed & Breakfast) offer the best value.