The transducer is included (with downscale, modified CHIRP, and standard), with the 1040. I believe no transducer for the 7610 is included--the transducers to utilize the full features of the 7610, can be on the expensive side, for side, down, dual CHIRP, and standard modes. That could bring the difference up to well over $1000--closer to $2000 with some transducers, but then you do have more sonar capability. The 7610 does have both 12 and 8 pin transducer connections, so you can run two transducers, and select--can run dual Freq CHIRP, down and side scan without an adaptor. The 1040 requires the "black box" for side scan (down scan is included)
I prefer the buttons when at speed or in rough water---but that may be a personal quirk (I have both touch and button Garmin screens on the current c Dory 22.) (1040 and 740S)
Touch leaves finger prints, which have to be cleaned off regularly. If you want video input--then the 7610 is the only choice between the two (or get a separate LCD screen for $300).
If you have a second station, there is an advantage to the daisy chained NMEA 2000 (two inputs). The 7610 does have both 8 and 12 pin depth sounder transducer connections.
To me it boiled down to touch vs button. I didn't have video input (nice but you can buy a LCD 7" Video screen for $300, The big difference is the transducers and side scan. If you want side scan, and the full 1000 watts, vs 600 watts, then get the 7610. Both excellent units.