You have the watches, but we have the time
Several sources I've checked attribute this quote to an Afghan proverb.
The meaning of the second part is clear: time is on our side. But what does the "watches" in the first part refer to?
Benjamin Harman's answer argues that the saying is a double entendre between "watch" as in "wristwatch" and "watch" as in "stand watch".
However, given that "watch" ("timepiece") and "watch" ("lookout") / "watch" ("keep guard") seem to be completely different words in both of the main Afghan languages: Dari and Pastho, it is not possible that the original Afghan saying (which I've been unable to find -- see comment) used the same word for all of those.
The most reasonable explanation is that "watches" here refers to the instruments, the hardware, the material means of winning a battle; while "time" refers to the passing of time and the immaterial means of winning: political changes, losing local support, difficulty of economically sustaining a war in the long time, etc.
EDIT: It turns out there's another widespread rendering of the saying which uses "clocks" instead of "watches". This reinforces the meaning of "watches" as actual timepieces.
The earliest match I've been able to find for any close variant of "You have the watches, but we have the time" is in testimony by Ambassador William Taylor, identified as "coordinator for Afghanistan, U.S. Department of State," in Hearing Before the Committee on International Relations, House of Representatives: Afghanistan Drugs and Terrorism and U.S. Security Policy (February 12, 2004):
Mr. TAYLOR. If I could just do one last comment on the Taliban. I have told this story once or twice before, but the Taliban, exactly as you have said Mr. Engel, have a saying that the Americans have the watches but we have the time. We need to prove them wrong. We need to be sure that they know that we are in this for the long term. Thy are not going to wait us out. We are going to be there, we are going to succeed. We are going to get this country, this government, this state on its feet so it can provide the services to its people and defend itself against the threat. Thank you.
Close observers of the U.S. political scene will recognize Mr. Taylor as one of the more impressive witnesses in the House inquiry into the Ukraine quid pro quo scandal that preceded the first impeachment of Donald Trump.
In any event, this earliest mention of the expression does not assert that it is an old Pashtun saying or an Afghan proverb, but rather that it is specific to the Taliban in relation to the war with the United States. A few later allusions to the expression attribute it to (different) particular individuals. For example, in a review of Seth Jones, In the Graveyard of Empires: America's War in Afghanistan in Choice: Publication of the Association of College and Research Libraries (2009) [combined snippets]:
At the end, he [Jones] quotes a Taliban detainee who told his US captors that "You may have the watches, but we have the time."
From Dominic Streatfeild, A History of the World Since 9/11 (2011):
As Mullah Omar famously stated: 'The Americans may have the clocks. But we have the time.'
And from William Remsen & Laura Tedesco, "US Cultural Diplomacy, Cultural Heritage Preservation and Development at the National Museum of Afghanistan in Kabul in Afghanistan," in Museums, Heritage and International Development (2015):
You may have the clocks, but we have the time. —Taliban spokesman
But more often, writers attribute the expression to "Taliban commanders," "a Taliban axiom," "a Taliban saying," or (more fancifully) "a Pashtun proverb," etc.
It seems clear from these various references that the "watches" alluded to in the expression are timepieces—with the metaphorical implication of superior technological hardware and scientific advantage (as walen's answer points out). The expression probably reached English from Afghanistan within the past twenty years, and it may well have arisen in Afghanistan within the same time frame.
Update (August 17, 2021): An earlier instance of the expression
In a further search of Google Books today, I came across this interesting instance of a closely related wording, from Richard Twiss, One Church Many Tribes: Following Jesus the Way God Made You (2000):
I was talking to my Maori friend Monte Ohia of New Zealand one afternoon about the way indigenous people think about time. He related the following story of his visit to South Africa among the Zulu people. He and his host were driving to a meeting whee Monte was to speak. When Monte asked what time the meeting was scheduled o start, his host said in about half an hour. When Monte asked how far away they were from the town where the meeting was to be held, his host said about an hour. When they arrived more than an hour late, all the people were waiting as though nothing were wrong. His host told Monte, "Don't worry about it. The White people have the clocks and the watches, but our people have the time."
This instance appeared in print four years before William Taylor used the expression in his Congressional testimony, and it differs from the Taliban instances in seeming to be about a difference in one's cultural conception of time, rather than an assertion about having time on one's side, as it were. Since it reaches us from a Native American author quoting a South African source by way of a New Zealand intermediary, it manages to circle about two-thirds of the globe without ever coming terribly close to Afghanistan. I am inclined to see it as a purely coincidental occurrence, but the similarity in wording is striking.