Witherhills they are both investigative. Treatment possible with both, but with Hysterosalpingogram the only thing they can do is unblock your tubes. With the lap and dye, you have a general anaesthetic. An incision is made in your belly button and they have a look around for any problems like ovarian cysts and endometriosis, and will treat if necessary. Dye is also inserted into your womb to see if your tubes are open and to check the general appearance of your uterus.
The Hysterosalpingogram is done without anaesthetic. Dye is inserted into your womb and an x-ray image taken of your womb/tubes. They can see if your womb looks ok and whether you have any blockages.
There's a pretty good medical explanation of both here: www.ekhuft.nhs.uk/patients-and-visitors/information-for-patients/a-z-of-patient-information/d/laparoscopy-and-dye-hydrotubation/#whatisalaparoscopy
I have had both done. With the lap and dye, it was performed by a v good gynae. No cysts or endo were found, though one tube was blocked (I had pelvic pain and was suffering infertility), however, he thought the lining of my uterus was not ideal so did a very light D+C. The results of that found that I had a small polyp, so my chances of conceiving were probably improved with that. He tried to unblock the tube, but was unsuccessful.
With the Hysterosalpingogram, I agreed reluctantly to have that done at the start of IVF investigation/treatment, despite it being only 8 months after I'd had the lap and dye. Consultant wanted to rule out water in the tubes, which would make implantation impossible. Of all the gynae investigations I've had done this was the worst. It was very undignified, and the gynae, though well-meaning, didn't take my history, so didn't know another consultant had tried to unblock my tube under general, so without warning he had a go during the dye procedure. I can tell you that was pretty painful and unsuccessful.
If you are offered a choice, I'd go for the lap and dye, though the general takes a good few days to recover from. If you have a good gynae, it's a bit of a fertility MOT and gives more investigative/treatment options.
Since the above, I've gone on to have a natural BFP after two failed IVF cycles this year.
Hope this helps! Good luck!