Jim Nguyen is correct about the outside influences. However, I found an additional Southern Chinese influence in Vietnamese cuisine, and I would suggest that both make similar use of noodles of various types and rice, and both use lots of soups.
The most significant difference is that the Thais use significant amounts of ferociously spicy chili peppers, both overall and in any given dish, while the Vietnamese use hot chilies sparingly.
Second, in Thai cuisine, the fish sauce is nowhere near as pungent, fishy, smoky and fragrant as Vietnamese nước mắm, which probably benefits from more extended fermentation.
The French and Southern Chinese influence on Vietnamese cuisine manifests itself in lighter or clearer soups and sauces, in which the tastes of principal ingredients is showcased. I find that the Vietnamese use rather more of a greater variety of sausages and patés.
I find that Thai dishes, while including excellent meats, seafoods and vegetables, strike more of a balance between featuring the main ingredients and emphasizing the sauce or broth.
The penetration of the two cuisines in the West is slow, in both cases, and uneven and incomplete in both cases. Restaurants offering excellent Vietnamese haute cuisine, in some geographic diversity, have begun to appear, going well beyond pho soups and bánh mì sandwiches. Thai restaurants usually offer significant variety, but still stick to the more conventional and recognizable ingredients, and are reluctant to unleash the full spiciness of most dishes. Thai ingredients are often found in mainstream American supermarkets, while even some basic Vietnamese items are only available in Asian stores