Partnership and a private limited company, both are small business organizations with a small capital and small working staff. I am not an expert in business and finance but based on my understanding and reading, the only difference I see between these two is while a partnership firm has limitations in terms of growing big whereas a private limited company can grow really big, sometimes even grow into the size of a public limited company.
A partnership firm is when you join with another person (can be more than one, and there is no limitation on how many people can participate, but if there are over a dozen it is generally recommended that you go for a public limited business model), raise capital, write a business contract, and register your firm in the trade and commerce department in the place of your operation. A Private limited company needs to be registered in the company registrar's office and need a document signed by shared holders.