First, stop using Rome2rio--for China, it can be wildly inaccurate and doesn't address all the nuances when you have to switch transport modes for a given journey. You cannot take a regular taxi from Beijing airport to Tianjin since it crosses jurisdictions and even if allowed, takes too long for cabbies to get back to the city to turn in their cab for shift changes. At night, it takes about an hour to get from PEK to Beijing South. There is an airport shuttle bus to Beijing South departing every 30 minutes until 22:30, it will take about 1.25 hours vs the direct taxi but costs only RMB 24.

You don't mention the critical factor: what time are you arriving into PEK? The options that you have are:
1) Intercity shuttle bus from PEK to Tianjin city, followed by Tianjin taxi from bus drop to your hotel/final destination. Buses leave every 45 minutes or from 19:00, hourly with last one is at 23:00, takes about 2.5 hours cost is RMB 82. Taxi cost in Tianjin varies with distance to final destination.
2) Taxi from PEK to Beijing South Station (approx RMB 120), followed by fast train to Tianjin, + taxi in Tianjin to final destination. Last fast train is at 23:05, takes 35 minutes, cost is RMB 66 1st class seat. In order to make this train, your international arrival into PEK would need to be around 20:00, to allow for Immigration/Bag Claim procedural time and transport time to Beijing South + ticket pickup.
3) Prearranged car/driver PEK to destination, door-to-door. This is going to cost a lot more than USD 60...probably more like USD 150 and will take 2.5-3 hours. If your flight arrival is scheduled for 22:00 or later, it is probably your only option if you must get to Tianjin that night.

You might consider just getting a hotel in Beijing for the night, then taking one of the early fast trains to Tianjin the next morning...they start running about 06:00. That would be my personal choice if arriving too late for reasonable public transport.