These terms have related meanings. Kafir is the broader term. Dhimmi is the legally invoked status of a Kafir.
A Kafir is a non-believer, an infidel, or an ingrate to the Islamic faith. A Dhimmi is alway a Kafir. But a Kafir is not always a Dhimmi.
From Wiki: In the Islamic doctrinal sense, the term refers to a person who does not recognize God (Allah) or the prophethood of Muhammad (i.e., any non-Muslim) or who hides, denies, or covers the "truth". The Quran also uses the word for Muslims; in Sura 2 Verse 256, it asks them to take upon themselves the action of "Kofr" of all unjust idols, persons or powers.
It is usually translated into English as "unbeliever" "ungrateful" or "obliterator." In recent times, the term is seen as derogatory, which is why some Muslim scholars discourage its use and suggest the term "non-Muslim" instead.[1]
A Dhimmi is a person who is a non-Muslim in a Muslim dominated society. Dhimmi is the subordinate legal status given to the kafir to "protect" him. Why is protection needed? Because Muslims are called upon to kill infidels as the general rule. The creation of the Dhimmi status was a way to offer protection to the infidel in exchange for fees or taxes, and a reduced level of cultural benefits or freedoms, like voting. If you are an infidel, it is a good idea to be a Dhimmi in a Muslim culture. Without being a Dhimmi you would be in violation of the rules, and the "other" option would be on the table, so to speak.
Views of modern Islamic scholars about Dhimmi (from Wiki)...
Ayatollah Khomeini in his book "On Islamic Government" indicates unequivocally that non-Muslims should be required to pay the poll tax, in return for which they would profit from the protection and services of the state; they would, however, be excluded from all participation in the political process.[146] Bernard Lewis remarks about Khomeini that his main grievance against the Shah was that his legislation allowed the theoretical possibility of non-Muslims exercising political or judicial authority over Muslims.[147]
Javed Ahmed Ghamidi writes in Mizan that certain directives of the Qur’an were specific only to the Prophet Muhammad against peoples of his times, besides other directives, the campaign involved asking the polytheists of Arabia for submission to Islam as a condition for exoneration and the others for jizya and submission to the political authority of the Muslims for exemption from death punishment and for military protection as the dhimmis of the Muslims. Therefore, after the Prophet and his companions, there is no concept in Islam obliging Muslims to wage war for propagation or implementation of Islam.[149][150]
Dr. Zakir Naik, a prominent Islamic preacher from India has stated that "as far as the matters of religion are concerned we know for sure that only Islam is the true religion in the eyes of God. In 3:85 it is mentioned that God will never accept any religion other than Islam. As far as the second question regarding building of churches or temples is concerned, how can we allow this when their religion is wrong? And when worship is also wrong? Thus we will surely not allow such wrong things in our (i.e. an Islamic) country."[152]
Dhimmitude is a more recently developed broader term implying those in a non-Islamic dominated society who are being intimidated or terrorized in some manner and are accordingly fearful of taking action to counteract their condition.
Bat Yeor's definition of "dhimmitude" (from Wiki):
"As for the concept of dhimmitude, it represents a behavior dictated by fear (terrorism), pacifism when aggressed, rather than resistance, servility because of cowardice and vulnerability. The origin of this concept is to be found in the condition of the Infidel people who submit to the Islamic rule without fighting in order to avoid the onslaught of jihad. By their peaceful surrender to the Islamic army, they obtained the security for their life, belongings and religion, but they had to accept a condition of inferiority, spoliation and humiliation. As they were forbidden to possess weapons and give testimony against a Muslim, they were put in a position of vulnerability and humility."[8]
Robert Spencer author of the The Myth of Islamic Tolerance defines "Dhimmi" (from Wiki):
... [Dhimmi] is the status that Islamic law, the Sharia, mandates for non-Muslims, primarily Jews and Christians. Dhimmis, “protected” or “guilty” people, are free to practice their religion in a Sharia regime, but are made subject to a number of humiliating regulations designed to enforce the Qur'an's command that they "feel themselves subdued" (Sura 9:29). This denial of equality of rights and dignity remains part of the Sharia, and, as such, are part of the legal superstructure that global jihadists are laboring through violence to restore everywhere in the Islamic world, and wish ultimately to impose on the entire human race. [10]
More here.