I'm a member of Historic Royal Palaces and I agree that there is plenty to see at both the Tower of London and at Hampton Court Palace, both for first timers and for returning visitors.
I'm not sure that I divide the Palace and the gardens so neatly for tourists and locals. I and my wife are nearly locals, it takes between 90 minutes and 2 hours to drive to either the Tower or Hampton Court Palace (we can drive to the Tower vicinity because of my wife's disability). Or about the same time for use to venture into London and get across town to either.
So I think I answer with a local's hat on, especially considering how many times we have been to both.
There is plenty to see inside Hampton Court Palace - the webpage does a very good job of laying it out and an excellent map. Both courts are very scenic, especially when you consider the clock, and Fountain Court has both the fountain and the - look up - sculptures. Seeing (and smelling) the royal kitchens is fun but takes little time. I have never been impressed by the wine cellar but I don't drink. Upstairs they have done a very good job of showing different private apartment areas in the style they would have been in. I particularly like seeing the Chapel Royal from above, where Henry VIII would have been, and then being able to be inside the Chapel Royal below and looking up. I especially like the ceilings, including the Great Hall and the Chapel Royal. When I go, I often just sit and contemplate. The wine fountain is one to contemplate nearby too, and they have added sound effects.
I've barely scratched the surface - there is really lots and lots to see.
The gardens are all different from each other and it will depend which ones interest you. There is the Great Vine of course, the hidden gardens, the formal gardens, the views into and from the Banqueting House, and views to the river, and then the very long Long Borders, the Wilderness, the Rose Garden (splendid in June), the Kitchen Garden, the Great Maze (oldest one around), the Park, the fountains.... there is even a horse drawn waggon which will take you into the further reaches.
No way that that can be all done in one day - I've been going for years and always find something new.
The little tuck shop is good for a meat pie and is inside, near the Chapel Royal, but it is small. The main café is open and airy and has very good food (not cheap but worth it) - everything from soups and salads to hot dishes to sandwiches to very tasty cakes and other sweets... There are lots of tables outside too, some on the patio and some on the lawn.
I have never felt the need to go off the property for lunch. In fact, since the lunch options at Kew Gardens are relatively few there have been many times that we visited Kew in the morning and stopped at Hampton Court for lunch (and a glimpse at the rose garden and kitchen gardens without going inside the Palace). An advantage to having membership at both...
I can try to answer specific questions, if I know the answer.
One piece of advice about timing - fewer crowds on weekdays, avoid Bank Holidays.