Current manycore processors exhibit large on-chip last-level caches that may reach sizes of 32MB - 128MB and incur high power/energy consumption. The emerging Multi-Level Cells (MLC) STT-RAM memory technology improves the capacity and energy efficiency issues of large-sized memory banks. However, MLC STT-RAM incurs non-negligible protection overhead to ensure reliable operations when compared to the Single-Level Cells (SLC) STT-RAM. In this paper, we propose an approximation-aware MLC STT-RAM cache architecture, which is partially-protected to restrict the reliability overhead and in turn leverages variable resilience characteristics of different applications for adaptively curtailing the protection overhead under a given error tolerance level. It thereby improves the energy-efficiency of the cache while meeting the reliability requirements. Our cache architecture is equipped with a latency-aware hardware module for double-error correction. To achieve high energy efficiency, approximation-aware read and write policies are proposed that perform approximate storage management while tolerating some errors bounded within the user-provided tolerance level. The architecture also facilitates runtime control on the quality of applications' results. We perform a case study on the next-generation advanced video encoding that exhibit memory-intensive functional blocks with varying resilience properties and support for parallelism. Experimental results demonstrate that our approximation-aware MLC STT-RAM based cache architecture can improve the energy efficiency compared to state-of-the-art fully-protected caches (7%-19%, on average), while incurring minimal quality penalties in the output (-0.219% to -0.426%, on average). Furthermore, our architecture supports complete error protection coverage for all cache data when processing non-resilient application. The hardware overhead to implement our approximation-aware management negligibly affects the energy efficiency (0.15%-1.3% of overhead) and the access latency (only 0.02%-1.56% of overhead).