Okay, so Hallstatt and La Tène weren't cultures as you might imagine them, but material cultures. This means that when you see someone describe something as "Halstatt", they are referring to a set of similar artistic designs, stylistic weapon shapes, jewelry, clothing and pottery that were found across central Europe during the early Iron Age, not a distinct culture with a single language, social structure, economic mode of production etc. It also refers to polities that existed within a cultural zone where the use of either material culture was widespread. Both correlate with proto-Celtic and Celtic speaking peoples living in central Europe but one did not have to speak a Celtic language to use the Halstatt material culture, nor did one have to use that material culture if they spoke a Celtic language.

So to answer your question more directly, Hallstatt material culture existed from about 1200-475/500 BC, and was mostly restricted to central Europe. Most of our examples of Hallstatt culture come from hillforts and princely tombs constructed in that period, probably by Celtic speakers, although Hallstatt culture probably extended to other linguistic groups, moreso than the La Tene culture would. Because the period of Hallstatt culture's existence is so long, archaeologists split it into four phases: A and B, which took place during the late Bronze Age, and C and D which were the transitional period between the Bronze and early Iron ages. During its initial stages, polities in the Hallstatt cultural zone were probably very small as archaeological evidence in the form of farmsteads and tiny villages do not suggest complex political organization, although burials of children with high status goods do suggest the hereditary transfer of status. Communities in the Hallstatt zone from periods A-C were clusters of farmsteads that roughly encircled a small village, tomb/religious sanctuary or a small hillfort. They would have been ruled by a chief living in one of the farmsteads, and disparity in wealth was not great. These communities were likely autonomous political entities theorized to have been between 7-15km in diameter.

By the C period (900-600 BC), things change completely. The number of fortified sites such as hillforts expands dramatically as a small fortification ruled by a hereditary aristocrat becomes the nucleus for nearly every community. Iron working becomes widespread (although most people doubtlessly continued to use bronze or even stone/wooden tools) and trade routes open up, with Mediterranean goods coming through the Alps via Italy, Baltic amber being traded southwards and salt mined at sites like Hallstatt being traded in all directions, which was likely the basis for the increase in importance and wealth of the hereditary aristocracy.

During the final D period of Hallstatt culture (600-450/500 BC), contact and trade with Greek and Etruscan polities becomes much more frequent, and will fundamentally change the political and social organization of the 'Celtic' world in central Europe. The establishment of Greek colonies in southern France moved the focus of political power from the Alpine region westwards, as polities closer to the supply of Greek commodities, increasing their political capital as Mediterranean goods were central in the Hallstatt prestige economy, dominated chiefdoms with unequal access to prestige goods. Because of this, centralized polities of an unprecedented scale for temperate Europe emerged, only to collapse and disintegrate around the 5th century. The first descriptions of Celtic peoples in Antiquity take place during this period as well, as the collapse of Hallstatt polities correlates with a period of massive Celtic expansion into the periphery of the Mediterranean world; it was during this period that Celtic peoples like the Boii and Senones invaded the Italian peninsula and settled in Cisalpine Gaul.

Increasing contact with the Mediterranean and the collapse of Hallstatt polities led to the development of the La Tene culture within the Hallstatt zone. This combined artistic elements of the the temperate European and Mediterranean worlds to create a material culture that is recognizably "Celtic". As well, the 'center' of this cultural zone shifted from the Alps to Gaul, but it was much more expansive than its predecessor, reaching from Ireland to Romania, and from Spain to Scotland. During the La Tene period political centralization increased rapidly; coins were minted in the La Tene cultural zone and quickly changed from being mere prestige goods collected for their intrinsic value and began circulating as currency, indicating the presence of political authorities that guaranteed their worth. Significantly large settlements (oppida) appear during the La Tene period and become fortified with ramparts relatively quickly. Oppida were either formed around existing settlements or consciously constructed and had their populations moved inside in a short amount of time. These sites can be qualified as towns and served as the center of political and economic power for the territory that surrounded them.

Arms and armour become significantly bigger and badder than those of the Hallstatt period, with the development of massive longswords, 4 foot long shields, chain mail and sophisticated helmets with neck-guards, as well as the four-pommel saddle which ended the use of chariots in warfare as cavalry became more dependable. Social stratification is much more evident than during the Hallstatt period, with society being largely controlled by a warrior-aristocracy and druidic caste drawn from that same aristocracy, the emergence of specialized craftsmen in oppida and the existence of a possibly substantial slave population. It is towards the end of the La Tene period that states actually emerge in temperate Europe, mostly in Gaul and southern England. In fact, Caesar describes many Gallic notables as magistrates elected yearly rather than hereditary kings, which may or may not be a simplification of Gallic political offices for Caesar's republican readership. The commercial use of coinage combined with written demographic documents described by Caesar indicate that a class of individuals endowed with administrative powers likely existed, at least in southern and central Gaul. By the time of the Caesarian conquest of Gaul beginning in 58 BC, many polities in the La Tene cultural zone had attained the qualifications of statehood, or were just on the cusp of becoming state societies. These claims aren't universal across the La Tene world and mostly characterize Gaul and southern England; the first indigenous state society in Ireland only emerged around the 12/13th century AD while coinage and urban settlements were only introduced there by Scandinavians in the 9th century AD.

La Tene culture sort of died out in continental Europe after the Caesarian conquest of Gaul but existed for some time under Roman rule. It did continue to exist in Ireland and Britain for several centuries, and continued to evolve with influence from Eastern Mediterranean monastic Christianity, Scandinavian artistic forms and so on.

I think /r/AskAnthropology might be the better subreddit for that question.