Bought our home for $970,000, sold it for $1,025,000 after owning it for a 1 year. After closing costs, our net is still $970,000. Is that still considered a gain?

that's a wash.

No Gain, no capital gains tax.

It is certainly  unlikely that the numbers will be identical, so you will have a small gain or loss to report.
Any gain is not eligible for exclusion.

A gain is reportable and potentially taxable. A loss (on a home) is not deductible.
 If you got a form 1099-S, the sale will have to be reported on your tax return, to avoid an IRS inquiry.