It's usually more visible by the vertical roadbed height, rail size, and ballast (or lack of it). You'll usually just see cinders and lighter rail. Usually the 'new' route got ballast lifts and new ties, the old one didn't; rail upgrades and the old one didn't.
Bridges are also a good indicator. Street running was a lot more typical on original railroads if the town got there before they did, and pretty soon, something had to be done to get out of that mess.
Curves would be last on the list, actually.
The award for the 'most miserable in-city curves that had to be bypassed' I've ever seen myself would be Lowell, MA, which dated back to the water and canal era, the railroads had to fight their way into the city via street ROW's. The current MBTA/Guilford main line is a pretty stark contrast to the now-trolley track that wanders around through the city through relatively impossible sharp curves and bridges.
Your original line very well may have been street running, too, which is an excellent reason to get out of Dodge and build a new high-speed bypass. I think Utica, Schenectady, maybe Rochester had situations like that with the original NYC main lines and subsequent bypasses. That's still being done today (Erie, PA didn't get NS out of 19th St. until the de-merger of Conrail and up onto the NYC right-of-way in 2001). Another classic is the old Delaware & Ulster coming up out of Rondout and heading west through Kingston, just a horrible combination of steep grades and sharp curves and street running that was never upgraded until the day it was removed. A small part remains as the New York Trolley Museum.
We had an 1880-built branch into our City by an NYC affiliate that got there late and resorted to street running over many blocks. They made a 90-degree cut through the middle of a residential block right through some houses. Always a 10mph curve and would just barely clear an 85-foot car. Measures out at a 500' radius curve which is still wide by model standards - but compared to the PRR main line it looked like a horsecar track.