How different were the "Spanish" people at the time of their colonization of Latin America to the Iberians at time of Roman Empire?
They Pelted Us With Rocks And Garbage
To start with during the Roman Empire they weren't really that Roman. It is far easier to forget all the nonsense you think you know about race, especially Celtic nonsense than correct it.
Not that different. It's hard to say, but I don't think there were drastic changes in less than 1,500 years. The genetic make up of modern Europeans is about 3-4,000 years old. They're descended from 3-4 groups, namely indigenous hunter-gatherers, neolithic farmers and indo-european peoples. Other groups (Siberians, North Africans...) made much less significant countributions. A study comparing mtDNA haplogroups found remarkable similarities between present-day Iberians and those of the Roman era. Unfortunately, studies like that are hard to do because ancient Iberians apparently cremated their dead. https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1529-8817.2005.00194.x
Genetically, very different. The ancestry of modern Iberians (comprising the Spanish and Portuguese) is consistent with the geographical situation of the Iberian Peninsula in the south-west corner of Europe. The large predominance of Y-Chromosome Haplogroup R1b, common throughout Western Europe, is the result of Central European invaders during the Bronze Age, making the Spanish and Portuguese population closely related to others from Western Europe. Similar to Sardinia and unlike the Balkans and Italy, Iberia was shielded from settlement from the Bosporus and Caucasus region by its western geographic location, and its low level of Western Asian admixture probably arrived during the Roman period. Later historical Eastern Mediterranean and Middle Eastern genetic contribution to the Iberia gene-pool was also significant, driven by Phoenicians, Greeks, Carthaginians, Jews and Levantine Arabs.
Hard to say, definitively. The Vandals passed through on the way to North Africa and the Visigoths spent a bit of time in the Iberian Peninsula. The Moors (Muslims) conquered and colonised the place in the 8th century and were not finally expelled until 1492. All of these groups stayed for a while and left genetic heritage.