A 5W-30 will generally provide marginally better fuel economy than a 10W-30, and certainly better cold startup protection than a 10W-30 in climates with a heavy winter. There are exceptions, where manufacturers post very close specs for their 2 oil grades. Many 5W-30s and 10W-30s from the same maker are chemically quite close, differing in the additive package and/or maybe the viscosity index improvers.
If you have a severe winter, you might want to use 5W-30 for the colder months and switch to 10W-30 for the wamer months. or just use 5W-30 year round. I don't deal with a winter worth speaking of, and I do have rather warm operating conditions in my area. 5W-30 is a broader "spread" than 10W-30, and my suspicion is that the 10W grade of the same oil - M1 in my case - will stay in grade longer and hold up better, longer than a 5W. This is at the expense of an MPG or so perhaps, but oh, well. I see no need to use a 5W in my area other than to "satisfy" CAFE standards, so I'm going 10W-30, which is still a listed grade and apropriate according to the temp charts in the manual. Your area may vary so check the chart for the grade(s) you can use.
If you do your own oil changes, keep your dated receipts so you can document them to a dealer in the future - may you never, ever need to though.