TLDR: the packaging makes next to no difference; even a small reduction in your cheese consumption will greatly outweigh the impact of the packaging (see the table further down)
Let's estimate the greenhouse gas emissions from cheese production and packaging. The data below relate to processing in different countries so are not directly comparable, but they should serve to give a first approximation. I'm focusing on greenhouse gases as arguably climate change is the most urgent environmental issue globally. Land use, water use, nitrification, other pollutants, biodiversity loss, etc are all also important (and are also all areas where dairy farming has high impacts)!
1kg of cottage cheese requires approx 6.5kg of milk to produce it (Journal of Food Protection Vol. 43. No. 6. Pages 441-446 (June. 1980) ).
1kg of milk production results in an average of 1.3kg CO2e emission (Greenhouse gas emissions on British dairy farms; DairyCo carbon footprinting study: Year one February 2012).
The production process produces a further approx 0.8kg CO2e/kg cheese (Tan, Aik Jong, "Energy Consumption and Greenhouse Gas Emissions from Commercial and Manufacturing Sectors Specific Studies on HVAC Equipment and Dairy Processing" (2012). Theses and Dissertations. 416)
So for the cheese, 1kg of cheese production results in 6.5 * 1.3 + 0.8 = 9.25kg CO2e.
Now for the packaging. Wrapping paper generates approx 1kgCO2e/kg of paper, and PP or PE bags generate approx. 2.7kgCO2e/kg of film (Methodology for assessing the climate change impacts of packaging optimisation under the Courtauld Commitment Phase 2, WRAP, December 2010). Let's assume 1kg of cheese takes about an A3-size piece of paper or film (0.125m^2) to wrap, the paper is 100gsm and the film is 100μm = also roughly 100gsm. Then we have 12.5g of packaging per kg of cheese, giving us (all data rounded to 2 significant figures as we're estimating):
CO2e from packaging CO2e from cheese total
paper-wrapped 0.013 9.3 9.3
plastic-wrapped 0.034 9.3 9.3
The question of whether paper or plastic is better all-round is very difficult to answer (and depends on things like where the wood pulp came from, what chemicals were used in the processing, how good your local recycling facilities are, whether it's going to landfill or incineration, etc, etc). But the vast majority of us shouldn't worry too much about that when we are still eating meat and dairy, still flying every year, and still driving cars around. Reducing your personal environmental impact dramatically is very simple, and can actually improve your quality of life (that's not to say it's not hard - I'm still working on this!):
- stop flying (or save up your money and holidays and go for one really long, really exciting trip every 5 or 10 years)
- insulate your home as much as possible (and look at heat pumps)
- cut out meat and dairy (or only eat really good quality very infrequently: you'll enjoy them more)
- stop driving (walk, cycle, use public transport; make changes to your lifestyle to make this possible - you'll be happier and healthier)
- stop buying stuff (mend, share, re-use, simplify - more life, less clutter)