So it's actually a bit hard to prove a negative (ie, why wasn't there an advanced ancient civilization around the Caspian), but here are some ideas to situate the Caspian Sea in the regions history.

Now there is a giant debate over whether the Caspian Sea is a sea or a lake under international law, and in many ways it has quite a few differences from the ocean: it has a third the salinity, overall, and it's an endorheic basin, meaning that in the recent geologic history it has no outflow or connection to other seas or oceans (it has a roundabout shipping connection via the Volga-Don Canal, which was opened in 1952). But even between different parts of the Caspian Sea there are vast differences. The southern part is the most "ocean like" in terms of salinity and depth (around a 1000 meters), while the northern part of the sea is much less saline (it's where most of the rivers emptying into it are) and is very shallow, with a depth of some 10 meters. The result of this is that the Northern Caspian often freezes over in the winter.

Despite this, it was of some significance, especially in terms of trade. It's just that the trading usage of the sea was mostly North-South, notably as the Kievan Rus' used it and the Volga as a means of trading with Persia.

Even as far as the Southern coast (including Gilan and Tabaristan) goes, it is a rich and fertile area that has long been of interest to Iranian states. It's exceptionally close to Tehran today, and is a traditional holiday spot for the city. It is cut off from the Iranian plateau by the Alborz Mountains, however, and therefore spent much of its history developing as a region with its own distinct languages and cultures. A similar geography and environment applies for the western shore South of the Caucasus: it is incredibly fertile, but the mountains mean that the natural flow is east-west in the Transcaucasus, or southwards towards Persia.

For the Eastern shore, this area is extremely arid and rocky, being on the edge of the Karakum Desert. The civilizations that did develop in the region were at oases along east-west trade routes, like Merv, or in more fertile river valleys, like along the Oxus/Amu Darya and in Khwarezm.

The northern shore is mostly in a steppe region, and the lower Volga has tended to be more a part of that region and its peoples than apart from it.

A lot of the human activity here is east-west, and also southeast-northwest: it's just easier to go by land around the northern Caspian shore to points westwards or in Russia than to go through a hassle of a boat ride across the sea. As such, the lower Volga river and delta tended to be used as a fertile base area for steppe empires. The Khazars had their capital at Atil, and the Mongols and Golden Horde had their capitals at Sarai. Despite the Astrakhan area being a fertile source of farmed fruits and vegetables today, however, there wasn't a particular need or level of separateness for a unique agricultural civilization to develop there: the steppe zone was well suited to nomadic pastoralism, and those nomads traded with civilizations South and with farmers in the forest zone just north, and upstream. The pre-Slavic upper Volga did develop under the Volga Bulgars, by the way.

So why didn't the Caspian Sea develop maritime traditions and civilizations? It is different enough in its parts, and separate enough from the Black Sea and oceans, that it didn't make a lot of sense. It has been a crossroads and has been used in trade for millennia, and has been of deep importance to the nomadic empires that rose and fell along its northern shores.