Simply supported design means you are designing reinforced concrete structure span by span. This type of design assumes that there is no transfer of moments between spans. For instance, in the design of a reinforced concrete frame structure consisting of 4 spans, using simply supported design method, you design each span separately.
The bending moment diagram for a reinforced concrete frame structure shown below consisting of 4 spans and designed using the simply supported method would look like this;
You should always use the ultimate load combination of (1.4gk + 1.6qk) in calculating design loading during analysis. Note that gk refers to dead load , while qk refers to live load.
Simply supported design is hand calculation friendly, you only need a scientific calculator to help you.
Simply supported design is a conservative type of design and in some cases may not produce an economic design.
Continuous design means you are analyzing and then designing structure as a whole. Since a reinforced concrete structure consist of slab, beams and columns, it means you are analyzing all together.
In continuous analysis and design, it is assumed that there is transfer of moments between spans or structural elements. Typical bending moment diagram for a structure consisting of 4 spans and designed as continuous would look like this;
Also in continuous analysis/ design, using only the ultimate load combination of 1.4gk +1.6qk sometimes do not give the result which can be described as worst case, various load combinations of dead, live and wind loads (where appropriate) need to be applied on the structure to determine the worst possible case under which the structure is safe.
While it is possible to analyze a structure as a whole using moment distribution, manually or by hand calculation, it is not practicable because it is tedious. Structural engineers who still design manually often divide the whole structure into sections consisting of 3 to 5 spans , then analyze and design each section separately until the whole structure is covered, and with the aid of structural analysis programs, the process has been made easier.
The use of computer in structural analysis and design cannot be over emphasized, computer can implement the use of moment distribution or finite element method to analyze and design a whole structure in minutes.