War and Peace is more grand in the sense that the background is the Napoleonic Wars. Everything feels very high stakes. There is a lot to keep track of, but it's extremely fulfilling in the end. A more challenging, addictive page turner (imo) with unforgettable characters and moments.

With Anna Karenina, there is more of a detailed emphasis on farming routines and peasant agricultural society to contrast to the glitzy urban setting of Anna's high society. It also feels more intimate, more modern / psychological, and easier to pick up and put down in a leisurely way (if you do this with War and Peace, you'll get lost). I'd call it the more 'beautiful' book.

In summary: W&P is the book that basically locked me in a room - I didn't put it down for weeks. Anna is the one that washed over me during a longer period of time. Since 'right book right time' really affects how much we enjoy something, I'd choose whatever sounds best and just skip to the other if you're finding either one a slog.