There is nothing wrong with neckties unless they are made of pure silk. Then, artificial silk is not a problem. Also, it a necktie contain insignificant amount of silk, like 20%, then it shall be considered permissible for men to wear.

The scholars allowed men to wear garments containing four fingers’ width of silk, based on the hadith you have cited in your question. Thus, it becomes crystal clear that only small amount of silken material in man’s clothing is permissible.
In this regard, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states: “There is almost a scholarly consensus that wearing clothes made of pure silk in the form of a shirt or tie, etc., is forbidden for men. Some scholars permit it in exceptional cases such as when the man has a skin disease that requires wearing silk, or some kind of dire necessity, or because nothing else can be found to wear, etc.
The above scholarly opinion concerning the prohibition of silk is based on the explicit statement of the Prophet (peace and blessings be upon him), “Wearing silk has been forbidden for the men of my Ummah.” The Prophet’s words are categorical in their implication, so there is no way to justify wearing pure silk in ordinary circumstances.
Concerning clothes that are not made of pure silk but are a mixture of silk and other substances such as cotton, acrylic, wool, etc., scholars are divided on the ruling about them. When one looks closely at the various views concerning this issue, it can be concluded that if silk is not the main substance of the cloth, then it is permissible; say for example: if a certain piece of cloth is 60% cotton and 40% silk, then it shall be considered permissible for men to wear.”