Llama is a family of language models whose weights are published, so they can be run on your own hardware instead of only through an API. It solves the problem that most capable models are reachable only as a hosted service, which rules them out when data cannot leave your network or when per-token cost dominates. It exists because releasing weights turned out to build an ecosystem of fine-tunes and tooling around one architecture, and it is the model most local inference tools target first.
DeepSeek is an open-weight model family whose published training methods attracted attention for reaching competitive quality at lower reported cost. It solves the cost question in training rather than in inference, which is where most optimisation effort goes. It exists as evidence that architecture and training-schedule choices matter enough to change what a given budget can produce, and its weights and methods are published.