Yes, the brand of charcoal does make a difference - especially when using it minion.
When using briquettes, generally the cheap briquettes you find in the supermarket are designed for quick grilling and the origin, mix and quality of the wood in the charcoal that is used to make them can be somewhat suspect. You really should be buying good quality kettle BBQ briquettes for long slow smokes. The difference between the two is that
the kettle briquettes are pressed to a much higher density and are usually made with predominantly hardwood charcoal. When you open the bag of kettle briquettes you should not see any significant briquette dust in the bottom of the pack. Drop one of each type of briquette on concrete and the cheap briquette will normally shatter whereas
the kettle briquette will normally bounce.
OK so there are physical differences - but what is the difference when cooking? The lower density of the cheap grilling briquettes means that they usually light quicker but they also burn a faster too. This does not help when you are trying to get a constant temperature over an extended period of time. The cheaper briquettes commonly produce significant black sweet (sickly) smelling smoke as they initially catch. This is usually due to the source and makeup of the wood that is used to produce them. When grilling this is not usually a problem as the smell goes as the briquettes reach temperature however when smoking using the minion method you are effectively continually lighting new briquettes over an extended period of time which will mean your meat is being continually exposed to the initial burn flavours of the briquettes.
Of course the cheaper briquettes do vary in quality and the Walmart brand may be fine. To see how good they are though, splash out on one bag of good quality kettle briquettes and a bag of your economy briquettes and light a chimney of each. Note the differences in smell and smoke density of each as each of the chimneys catch. I think you may be surprised at the difference.