Of course there is a difference; one is the thing itself the other is the measurement of that thing. In this case time is duration and sequence and the clock produces a set of events in a particular sequence with a standardized duration between them. If something occurs that effects the duration between those events then the measurement will be effected but not the actual amount of time itself that has passed, even for the effected clock.
If the two clocks in Bob's hypothetical had spent their entire existence within a line of sight of one another, say one was perched atop a 1 mile high tower* directly above the other, then each could have observed the other the entire time and while one clock was running slower then the other, neither would have ever left the others present. As Bob put it, they would both have observed the same number of sunrises, sunsets and any other cosmological events you want to include like new moons or shooting stars, comets etc. In fact, if you were to extend the exaggerated nature of Bob's example to the point where the clocks read a full years difference in elapsed time, then one clock would be, according to Einstein, one year younger than the other but the same number of days old, which is, of course, a contradiction and therefore impossible.
Resting in Him,
Clete
* Note also that the top of such a tower itself would be “traveling through time” at a different rate than the base and yet would remain intact in spite of this and so now you would have a single object in two different space-time continuums simultaneously.