I tend to agree with Rocky's answer. However it sounds like you want to look at this from the numbers side of things. So let's consider some numbers:
Cost of new car: $25K
Value of old car: $8K
Out of pocket expense: $17K
I'm assuming you have the money to buy the new car available as cash in hand, and that if you don't buy the car, you'll invest it reasonably. So if you buy the new car today, you're $17K out of pocket. Let's look at some scenarios and compare. Assuming:
Annual Inflation Rate: 2.5%
Old Car Depreciation Rate: 8%
New Car Depreciation Rate: 20%
Annual Investment Returns: 5%
If you buy the new car today, then after 1 year you'll have:
Starting Cash: $17K
Value of Old Car: N/A (sold)
Cost of New Car: $25K
Value of New Car: $20K
Remaining Cash: $0 (spent buying new car)
Total Assets (new car + cash): $20000
If you keep the old car, after 1 year you get:
Starting Cash: $17K
Cost of Tires: $1K
Value of Old Car: $7360
Cost of New Car: $25625
Remaining Cash: $16800
Total Assets (old car + cash): $24160
After 2 years, you have:
Starting Cash: $17K
Cost of Tires: $1K
Value of Old Car: $6770
Cost of New Car: $26265
Remaining Cash: $17640
Total Assets (old car + cash): $24410
And after 3 years, you're at:
Starting Cash: $17K
Cost of Tires: $1K
Value of Old Car: $6230
Cost of New Car: $26920
Remaining Cash: $18520
Total Assets (old car + cash): $24750
Or in other words, nothing depletes the value of your assets faster than buying the new car. After 1 year, you've essentially lost $5K to depreciation.
However, over the short term the immediate cost of the tires combined with the continued depreciation of the old car do reduce your purchasing power somewhat (you won't be able to muster $25K towards a new car without chipping in a bit of extra cash), and inflation will tend to drive the cost of the new car up as time goes on. So the relative gap between the value of your assets and the cost of the new car tends to increase, though it stays well below the $5k that you lose to depreciation if you buy the new car immediately. Which is something that you could potentially spin to support whichever side you prefer, I suppose.
Though note that I've made some fairly pessimistic assumptions. In particular, the current U.S. inflation rate is under 1%, and a new car may depreciate by as much as 25% in the first year while older cars may depreciate by less than the 8% assumed. And I selected the cheapest new car price cited, and didn't credit the tires with adding any value to your old car. Each of those aspects tends to make continuing to drive the older car a better option than buying the new one.