Although they are completely different concepts, they are certainly quite united the majority of the time, and one is often representative of the other in no particular order.
Linguistics is a field of study that focuses on language, its formation, origin, structure and use. It is strictly a constant analysis of how language is broken down, how it changes over time, and what are its main units or components. This includes everything, from the way words are formed individually, to what role each word plays within a sentence, to what context is used to define the term, and even the sounds and symbols that bring words to life.
Culture is a human composite of behaviors, prerogatives, systems of belief, ethnicities, and other factors that distinguish one group from another group.
Language is one of the identifying factors of culture. This is what linguistics and culture have in common.
How does language identify culture, and culture identify language?
1. The way we say things: Prosody, which is the emotional intonation of words when we communicate, is used differently from one culture to another. The whims and whimpers of one group sound entirely different to those of another group because of what Albert Bandura calls "Social Learning". When applied to linguistics and language acquisition, this is a process of mimicry where humans tend to adopt the speech patterns and word choices of the majority. Therefore, it is no surprise that a child raised in a country that is not his or her own will develop the manners of expression, the cadence, timbre, and pause of the country where he or she will group up.
2. Culture can also be defined by geographical location the same way that linguistics traits are. If it wasn't for this, we wouldn't have regions in the world united by language or dialects. Therefore, there is a cohesive relationship between these three things: geography-culture-language.
3. Culture and linguistics are based on input. Contrary to wrong thinking, your genetic makeup does not entirely prescribe your emotional and psychological make up. All factors involved in personality are not all inherited. According to Gordon Allport's famous personality theories, we all have primary and secondary traits which stem from our environment, as well. Therefore, we can be conditioned to speak, act, feel, and identify with cultural groups and languages whether we are born in the nest of them or not. Input, external particularly, defines us just as strongly culturally and linguistically.
4. Culture and linguistic traits can be supplanted by other traits when the original ones are not used. For example, it is known that not using a language makes us forget it. Similarly, moving away from cultural practices, cultural ideals, and cultural identity will also render us prone to move away from our original cultural/linguistic make up. In other words, they are unfortunately replaceable.