Congrats on the two offers! You are very fortunate.
Aside from the criteria you mention, I'd heavily recommend you also think about location, and how that is aligned with your long-term goals. It's a big difference between ME and London.
Now, let me take your questions one by one.
1. Type of projects: My preference would be more finance / core strategy projects and less operations / execution projects
You can expect OC&C to do more of the former and Strategy& to do more of a latter. This is based more on my perception and knowledge about the firm rather than official data. Still, it's hard to tell anyway because the project type mix also differs from geography to geography and it's also something that you as a consultant in that firm have an influence on.
2. Conversion chances from Internship to full time
Strategy& is larger and thus more able to absorb more people. Still, don't take the decision based on this. Your likelihood of converting it into a full-time offer has more to do with your performance during the internship and that in turn is influenced by how much you enjoy the company culture and type of projects.
3. Work culture
Overall, not very different. The differs regions will have a big impact though on the mix of people you're going to find in that particular office.
4. Growth potential
Similar. Perhaps with Strategy& you have more flexibility since it's a larger company.
Best of luck!