Yes, you will get different p-values: limma uses a different approach to assessing statistical significance than the ordinary Student's t-test.
And you need to use the adj p-val (to correct for multiple comparisons; since microarrays contains thousands of probes, to determine which ones give statistically significance differences in differential gene expression analysis, you need the adjusted p-vals). If your goal was to just look at the expression of a single gene, then no need to do multiple comparisons correction (but if you're looking at just one gene, you wouldn't be using microarrays anyway).
For making graphs, I'd recommend just plotting log fold changes for each gene rather than plotting two bars: average expression of treatment samples for gene X and average expression of control samples for gene X. But regardless, in either case, use the limma adjusted p-values.
I suggest you read the limma papers to understand what it is doing (model fitting etc). In short: It is not doing a simple t-test as dsull already pointed out. Run the entire dataset through limma and take the adjusted p-values. Do not start custom approaches outside of the established packages for RNA-seq if you have no expert knowledge. The introduction of this manual cites the relevant literature: https://www.bioconductor.org/packages/release/bioc/vignettes/limma/inst/doc/usersguide.pdf