I would say that there are two main conflicts in the story. The first conflict is a simple "who is right" argument. Is it better to punish someone by capital punishment or is it better to do life in prison? The lawyer believes that prison is better; the banker believes that death is better. In comes the bet. The banker puts up 2 million rubles and the lawyer agrees to stay under house arrest for 15 years.
Once the time in prison begins, the conflict switches to an internal conflict for both characters. For most of the 15 years, the internal conflict is centered squarely on the lawyer. He's definitely struggling with solitary confinement, and what makes it worse is knowing that he can walk out at any time (but lose the bet). By the end he is at peace with the prison sentence but in conflict with everything that he has learned about humans and society. He's disgusted with all of it and intentionally loses the bet with 5 minutes to go.
The other internal conflict is the conflict raging inside of the banker as the bet nears its conclusion. The banker is almost completely broke. Paying the 2 million rubles will wipe him out. The banker decides to kill the lawyer and frame somebody for the act. He's wrestling with knowing how wrong that is and knowing that he really wants to keep his money.
Chekhov makes it clear that by the end of story, both men have been utterly broken by their own internal conflicts.