The Germans were generally happy with the Me-190. However, the day would come when newer and better fighters were in service in enemy air forces, and they wanted to be ready for that day. There is usually a significant lag - many years - between the initial design work, first flight, and operational service. For example, work on the Me-109 began in 1933, it first flew in 1935, and went into service in 1937. If a new enemy fighter that outclasses your fighters appears in the skies, you want to have started work on your new fighter years before that time. This was the purpose of the Fw-190: to be a potential successor to the Me-109. The search for the next fighter began in 1937, with the Me-109 just entering operational service.
The Fw-190 had a similar lag between early design and operational service - first flight was in 1939, and it entered service in 1941. Some Germans advocated replacing the Me-109 with the Fw-190 (Kurt Tank, the designer of the Fw-190 was one, but he had obvious personal motives), but both aircraft were used by the Luftwaffe to the end of the war. There were three reasons for this:
The Fw-190 was not completely superior to the Me-109. The Me-109 had better high-altitude performance, while the Fw-190 had much better low-altitude performance. This meant that the Me-109 was better for high-altitude combat against enemy fighters.
Improved versions of the Me-109 were developed and went into service.
German industry had limited capacity. This helped with the selection of the Fw-190, which used a radial engine. If it had used the same engine as the Me-109, Me-109 and Fw-190 production would have competed for the limited supply of engines. A completely different engine avoided this. However, this also made it harder to convert from production of the Me-109 to the Fw-190.
Some manufacturers who were making the Me-109 switched their production lines to the Fw-190. However, new factories were built to increase production of the Me-109. The Me-109 also benefited from lower production costs as a result of some of the wartime redesigns, and late-war models of the Me-109 appear to have had lower production costs (e.g., measured in terms of man-hours required) than the Fw-190. Even though inferior in some ways, it had become cheaper, and continued production let German have more operational fighters. Even without considering the sometimes-convoluted politics and favouritism in the Nazi state, there were good pragmatic reasons to keep producing the ME-109.