Generally speaking...
FIX messages are relatively longer than API messages.
In this case, you will be using 35 = X (MarketData-IncrementalRefresh). Size is usually 200 or so. A usual TT-API Quote message is about 1/2 - 3/4 of a typical FIX msg.
That said, FIX is not the answer to faster quotes or order execution. What FIX offers is fully custom control (At least, to a level where the broker allows it) over how you want your messages sent to your corresponding business. It also gives you an easy change of implementation when dealing with multiple brokers.
In this time of age, latency is mostly due to the user side's ISP setup and workload depends on how the initiating side's programmed the FIX Engine and it's surroundings.
FIX is a msg. exchange specification which banks and brokers agree to communicate under. It's a bit of an hastle learning an API semantics and rebuilding your program everytime you change brokers. By using FIX, you can connect to other (multiple) brokers with just a few configurations and setup. Disregarding the "Test-to-Production" Tests you go through the brokers/ banks, most with FIX experience can get their application connected to a new broker in just a few hours or minutes.