I first felt old when the centerfolds could be my daughters. Then, when Joe Strummer died, that was another seminal moment.
OK, to the point - when did feel old as regards to running. In my youth, I was decent enough to train with a group that put four guys on Olympic teams. Mid 14s for 5K for me was my best at age 25. Very intense training - lot of speed, little mileage. Ran for a shoe company and all that, then I went out to the real world.
I was still running upper 15's for 5K into my work life until age 33 - then low 16s until 36. At 37 I started to slow & retired for 4 years.
Got back into it at age 41 4 years ago - was able to shed 40 pounds, but am 26 pounds over youth racing weight of 146. Might get to about 165, but not that important anymore.
I can train injury free with moderate intensity about 40-50 miles per week. I've had gusts to 17:26 as a master, but I'm consistently low to mid 18s for 5K. Running over 50 miles per week tends to get me hurt - sadly most of my running is on roads due to where I live, I think I could run more on softer surfaces. I've been fortunate to be consistent over the last 3 years - not getting faster, but not that much slower.
Main differences in getting older have been well documented by other posters - delay in recovery is the most notable - particularly from nagging injuries. Little things can take you out for a week, instead of an afternoon. Ice is a big friend. Stetching has to be done after every run or I feel it the next day.
Top end speed goes away quickly, I do a lot of 30 30/30 sessions to try to get some of that speed back while not letting the mileage suffer. I'm now in my fall buildup - every four days I go to run halves or so (done on the road since the local track is an absolute brick and wearing spikes hurts my calves). Can still do 6 halfs at 2:35 - but now I take a full 3:00 recovery (in youth I was doing 2:05-10 halfs on a 4:30 cycle at the beginning of the year, getting down to a 3:30 cycle at top form). So, it's an adjustment, but not unacceptable.
Frustration is the biggest thing. I'll go to a local 5K and know in my prime I could beat anyone there - and in my head I still know how to race. It's just the body which can't do what it used to. The guys who used to run away from me could run 13s for 5K, not 16s. Hard to wrap the head around, but you keep going.
Motivation was a big response from other posters. I'm motivated in different ways now - my big race is in November, and I try to be within 1% per year of my best time in my youth. So, now I'm 20 years older - I want to be within 20% of my best time. I've made it the last 3 years and should make it again. The occasional award is nice - the rare win is an event - but mostly, if you enjoy competing and testing yourself, there's a reason to keep training even at a non all-encompassing manner.
You know how when you're in a mile race and you get that lactic buildup in the last 200 - but that's ok because you're all out? I ran one hard mile this year - went out in 75 and had that same buildup - with 1200 to go. That's what it's like getting old.
My father started running in his 40s - was able to run a 3:08 marathon at age 50. At age 66 he started slowing dramatically - from 22:00 5Ks to 24, then 28, then 30, and now 33:00 at age 72. His lesson, don't take time off. Everytime he goes on vacation with my mom (spending my inheritance he calls it) he misses 10 days - and comes back slower & slower.
On the other hand, I went to my 20th college reunion and won the award for looking most like my younger self. To this day the ladies think I'm mid 30s because I'm still trim and have youthful skin from the running. I still have my hair, but running has nothing to do with that. So, yes, you're going to get slower - that's old age. But there are benefits to continuing that don't show up on the clock.