The most important difference was chronological one, as NKVD and KGB never existed in the same time under such names. Bu the history of the Soviet organizations responsible for internal affairs was quite turbulent and might require additional explanation.

NKVD (People Commissariat of Internal Affairs)has been formed in 1917, shortly after the Revolution and was initially devoted to manage administration and civic order, overseeing the functioning of civilian administration services, police force (Peasant and Worker's Militia), fire departments, prisons, labour camps and border guard. Initially, the role of special service merging the function of secret service, intelligence agency and political police was assigned to People's Commissars Council All-Russian Commission for Combating Counterrevolution and Sabotage created on 7th December 1917 OC and renamed People's Commissars Council All-Russian Commission for Combating Counterrevolution, Speculation and Power Abuse in August 1918. Due to rather long name, it was most commonly known as VChK (All-Russian Special Commission) or simply 'cheka'. This service was officially tasked with investigation and policing counter-revolutionary activities, but in reality it was a service responsible for internal security, intelligence and counterintelligence (both civilian and military) as well as the military part of the border control. Due to VChK's infamously brutal actions during the Civil War and the post-Revolution wave of political terror, VChK has been officially disbanded on 28th December 1922 OC and replaced with GPU (State Political Directorate), although the change was only symbolic as GPU retained all the powers of its predecessor. The service was again renamed to OGPU (United State Political Directorate) in November 1923 after the creation of USSR. Nominally, OGPU was reporting to NKVD.

This changed in 1934, as Staling began his preparations to the Great Purge. After the death of Viacheslav Mienzhynski, Director of OGPU, the service has been substantially reformed and as GUGB (Main Directorate of State Security) and became one of the Directorates of NKVD, meaning that the latter gained an immense power controlling both the entirety of internal affairs as well as being a (counter)intelligence agency with substantial military reserves.

Another change came on 14th April 1943 with the act issued by Presidium of the Supreme Soviet that created NKGB (People's Commissariat of State Security) formed of the 1st-4th Directorates, Directorates for Transport and Economy (essentially, intelligence and counterintelligence) and 1st, 2nd, 3rd and 5th Special Units (special police operations) of NKVD, mimicking the initial division of duties from 1917, when the intelligence and special operations were carried out by an agency outside NKVD itself. In 1946, on Stalin's orders, People's Commissariats were reorganized and renamed as Ministries. Thus, NKVD has been renamed MVD (Ministry of Internal Affairs) and was headed by Sergey Kruglov while NKGB was christened MGB (Ministry of State Security) with Vsevolod Merkulov as the corresponding minister.

After the death of Stalin, both MGB and MVD fell under strong influence of Lavrentiy Beria who managed to merge the two into MGB and securing the position of this ministry, establishing firm control over the entire apparatus of the internal affairs. This was most likely the main reason behind the coup orchestrated by Khrushchev and Zhukov as the influence of of Beria could have destabilized the balance of power maintained between the three pillars of the Soviet government - Political Bureau, internal services and the military. After the imprisonment and execution of Beria in December 1953, the MGB was divided again. Most of the former units of the MGB and some units of the pre-merge MVD were separated to create a new organisation, KGB (Committee of State Security). Despite the name, it was equal in rank to all ministries, reporting to the Prime Minister directly. The first president of KGB was Ivan Sierov who resigned in 1958 to stand at the helm of GRU (Main Intelligence Directorate), the military intelligence agency. The KGB itself remained the organization responsible for foreign and domestic intelligence and counterintelligence including the function of political police as well as protection of all affairs related to governance. In 1991 it has been disbanded and its specific units were either transferred to MVD or reformed into new organizations, such as FSB (Federal Security Service) or SVR (Foreign Intelligence Service).

So to sum it up: NKVD (later MVD) was an equivalent to a Ministry of Internal Affairs in the years 1917-1934, 1943-1953 and 1954 onward (it exists today), having received the prerogatives of intelligence agency and secret police after merges with respective agencies in the periods of 1934-1943 and for few months of 1953-1954. KGB was an intelligence agency and secret police formed in 1954 and operating continuously until 1991, an indirect successor of the organizations known as VChK (1917-1922), GPU (1922-1923), OGPU (1923-1934), GUGB (1934), NKGB (1943-1946) and MGB (1946-1953).

EDIT: Minor spelling and stylistic issues.