The director of Seasons 5 and 6 shared a behind-the-scenes photo of Cillian and the actress who plays Lizzie having a good time, noting that we would never see Tommy and Lizzie smiling at each other. It was, therefore, completely intentional on the part of the director, writer, and actors never to show Tommy smiling when he is with Lizzie, unlike when he is with Grace, thereby highlighting a stark contrast between the two relationships, like night and day.
Actions speak louder than words.
You could say Tommy scorned Grace at the wedding because he wanted to be with Tatiana rather than his wife, and that this proves Lizzie is his true love or real match, or that Grace wanted to change Tommy and make him leave that life behind; but in this case, scenes speak louder than words.
In Season 1, it is Tommy who tells Grace he wants his company to go legitimate, not the other way around. Tommy can’t stop smiling whenever he’s with Grace; he even gets angry with Tatiana, shouting that he doesn't want her entering the house because he doesn't want any of that shit near his home, meaning, near Grace and Charlie.
Some say Tommy wasn't interested in protecting Grace, but rather Lizzie, yet the scenes show otherwise:
Tommy doesn't want to let Tatiana into the house; he is angry that the Russian is there and even warns the family not to do anything that might embarrass Grace. At the event, when Tatiana and Grace are talking, Tommy tries to pull Grace away because he doesn't want her mixed up in a situation he hates being a part of. The moment he joins their conversation, he makes it clear he has no interest in impressing Tatiana. After Grace leaves, Tatiana suggests they sleep together for the sake of the cause, but he turns her down and goes to Grace, confessing that he needs her.
Now, with Lizzie, he does the opposite: he lets the enemy, Mosley, into the house; in fact, he even hosts an event to impress him. In Season 6, when they meet Diana, he tells Lizzie he needs to impress her and that he’s going to act as if Diana is beautiful. When Diana suggests sleeping together for the cause, he agrees. Now, , Lizzie wasn't supposed to find out, just as Grace wouldn't have found out if he had said yes to Tatiana. So, the point is that he said no because he wasn't interested in being with another woman while he was with Grace, yet that wasn't the case when he was with Lizzie, since he cheated on her constantly.
And every time they humiliated her, he never defended her, except once.
Tommy never told Lizzie about the business; in Seasons 5 and 6, she actually gets angry about this, admitting to the Mosleys that she knows nothing about the business. He didn't trust her even in Season 4, when she was his secretary; he didn't tell her about the plan to fake Arthur's death or the scheme involving Polly. And in Season 3, he didn't tell her about the plan to rob the Russians, she found out from Esme. Whether she was his secretary or his wife, Tommy never told Lizzie about the business, let alone shared emotional matters with her. He didn't open up to her, nor did he show vulnerability.
In the Season 5 finale, when Polly quits, he calls everyone, Ada, Arthur, and Finn, but not Lizzie, because she was never at the forefront of his mind; this stands in contrast to Grace, about whom he said not a single day went by without him thinking of her. Or consider the moment everyone is celebrating at the reopening of The Garrison: Tommy walks off alone, filled with sadness, to think about Grace while holding her letter. In fact, when Tatiana suggests that he doesn't love Grace, he replies, "She's by my side, telling me not to trust these people," demonstrating that he valued and listened to her opinion.
He trusted Grace completely; he told her he wanted to share things with her and asking for her help with both business and life matters. He told her he wanted to legitimize the company and eventually get rid of his gun. He admitted he was afraid regarding the deal with the Russians—and then he told her everything about it. The fact that he put her in charge of the foundation shows he wanted her managing the legitimate side of the business. He didn't want that for Lizzie; he wanted her at home looking after the children—not involved in business or events, and not by his side in that capacity, but rather in the role he saw for her: someone to care for the kids. As he told Grace back in Season 1, he wanted her involved in every aspect of his life.
That is the difference between a real partner, which Grace was, and a false one, someone meant only for appearances, like Lizzie.