An often-misunderstood aspect of credit score reporting is that the score can vary based on two dimensions: The bureau it's pulled from, and the scoring model used. In other words, not all Experian scores are the same.
You've identified Experian in your question. They are one of several major credit bureaus used in the United States. Their job is to collect information on consumers and calculate credit scores, then sell access to those scores to financial institutions or the consumers themselves. The first potential variance is that each bureau does this collection-scoring-distribution process a little differently, and they may have different data available to them, or they may receive data in different timeframes. So, a score from Experian may be different than a score from another bureau.
However, that doesn't seem to be the case for you, since you've mentioned that both scores came from Experian. In this case, we can assume that the difference was because a different model was used. The data a credit bureau collects on you is fed into a mathematical model in order to determine the score. The model is what assigns points to different factors (i.e. age of credit, utilization, late payments, and so on). There are several scoring systems in use, but the major systems are FICO and VantageScore. Within each of those systems, there are different versions of the models (so, even two FICO scores may be slightly different, depending on the FICO version in use).
CreditKarma uses VantageScore. Most financial institutions who are evaluating you for lending will use one of the FICO scores. This can mean that your CreditKarma score is different than the score a lender sees when they pull your credit. Note that it doesn't inherently mean that either score is wrong - it just means that they're calculated differently.
To bring this back to your example, you said:
The approval letter included her Experian score, which stated a score of 645
That's not really a full statement - likely, her Experian FICO score was 645. And it was different from the 700 score she saw on CreditKarma because that was her Experian VantageScore.
You also said,
we don't know credit well enough to figure out what the issue could be
That's easy to solve, as there's lots of google-able information on the web that explains credit models and how they work. Also, since you're already using CreditKarma, do some digging in there - they provide educational tools and even specific feedback on why your score changed each month. It's also worth following up with her bank, as many financial institutions are happy to explain how they evaluate customers for lending, and what factors are actually important (hint: credit score is one of many, and often not even the most important). In the end, now is a great time to start learning about what influences lending decisions, since it'll help you change your behaviors as needed to make sure you're set up for any future lending plans (i.e. a mortgage or other major loan).