To provide a full understanding of your question, it may be helpful to get some context on the different parties involved.
Consumers like you and me utilize credit accounts and take other actions that create a credit history. The credit accounts may be at banks, credit unions, or other lenders. These lenders have contract arrangements with credit bureaus wherein they report the status of their customer's accounts regularly.
The credit bureaus collect information from each lender and tie it to consumers to create a credit report for each consumer. The report is essentially the detailed contents of your history - it shows accounts, balances, and so on.
Lenders, consumers, and other parties can request this credit report data from a credit bureau for different purposes - for informational purposes (a soft pull) or to establish credit-worthiness (a hard pull) in order to open a new account or perform some other financial activity. The information included in the report is the same regardless of why it's pulled, in other words the information Credit Karma pulls on your behalf is the same information as is pulled by your bank when you apply for a new mortgage.
Banks or other institutions typically have their own underwriting criteria to help them approve or deny applications. This criteria typically makes use of many factors included in a credit report - for instance, the debt half of a consumer's debt to income ratio is typically calculated by adding up all the monthly payments for open accounts listed on the consumer's credit report. (Notably, credit reports don't include income information, so the income half of the equation is obtained via other means).
There are three major credit bureaus, and typically lenders will report data to all three - although there are cases where some data only makes it to some bureaus, so it's often the case that if you request reports from each bureau and comb through the details, there will be differences. For instance, last year, I had my credit pulled as part of establishing a new contract with a propane vendor. It shows on my Experian report but not on my Equifax or Transunion reports.
Perhaps the most famous element of a credit report is the credit score. Credit scores were essentially invented to provide an arbitrary scale to measure risk. The intention of a credit score is to indicate the likelihood that a consumer may default on a loan in the near future.
Of course, that's a hard thing to predict, and different analyses result in different answers to that question. This has lead to the (often confusing) situation where credit scores may differ wildly, even when requested from the same bureau on the same day. Essentially, there are two major providers of scoring models - FICO and VantageScore Solutions (which is really just a conglomerate formed by the three major bureaus). Each of these scoring models is available in different versions, and new versions are released regularly. For the most part, the most current models from these two providers consider the same types of factors, but they weigh them differently.
To bring this back to your questions, you painted a scenario:
I consent to Business/Person A to run a credit report on me.
Person A runs a credit report on me. They get Credit Report A.
I go to credit karma to get my credit report, we'll call this Credit Report B.
Is there a material difference between Credit Report A and Credit Report B?
Further, you clarified,
I'm interested in the content of the credit report about my credit worthiness, like if there's different personal information or credit history information between the reports or if they are identical.
There may be many material differences between the reports in terms of both the contents and how they are used to determine credit-worthiness. Firstly, a report is pulled by request from a specific bureau (although some services like CreditKarma pull from multiple bureaus by default). If Report A and Report B were pulled from differnt bureaus, there may be material differences between the two in terms of one containing things the other doesn't contain (my propane example above).
The second typical difference is the credit score. Besides pulling a report from a specific bureau, entities requesting credit reports can specify a specific scoring model.
This is important when services like CreditKarma are mentioned because they typically pull VantageScore-based scores, while many (most?) lenders use FICO to make actual lending decisions. This often leads to the case where a consumer is sitting in front of a loan officer showing them their phone, where the CreditKarma app says their credit score is X, while the loan officer looking at the bank's lending system is seeing a credit score of Y.
But even within the same brand of scoring model, there can be differences from version to version. For instance, if Report A was pulled with FICO 8 and Report B was pulled with FICO 9, and there were medical collections accounts reported for that consumer, the score pulled on FICO 9 will be slightly higher.
