The differences are in naming; those differences exist across firms (and less frequently across regions). Despite naming, career path, responsibilities and qualifications required at comparable roles are the same across firms/countries.
Below, a quick recap on differences in naming (depending on firm) and their responsabilities:
Analyst (or Business Analysts, BA) (McK and ATK) / Associate (BCG and Bain): this is the entry level role for recent graduates or candidates with little work experience. They are with the firm for up to 3 years. Their role is similar to associate (or consultant depending on naming) but they work less autonomously. Associate (or consultant depending on naming) or managers often help guide their work, spending extra time to discuss what to analyze, how to analyze it and how to put it on slide. They usually work on particular parts of a problem and supply their output (usually an analysis and a PowerPoint) to the associate (or consultant depending on naming) or manager.
Associate (McK and ATK) / Consultant (BCG and Bain): are with the firm for up to 4/5 years. Post-experience or MBA position. They do the major chunk of the analytical work and content production. It is them who actually drive the engagement towards a recommendation and implementation. At this stage, they are highly independent and need to drive their work streams autonomously. The project manager and partner will come to them for their opinion and advice on particular matters. They are in close contact with the client team every day. Depending on their tenure, they are asked to manage parts of the project.
I hope this helps,