Taste wise some versions can be very similar.
Technically, they're quite different in many ways. That can have different malt, hops and yeast profiles and combinations. And I haven't even touched on the differences between Amber ales and Brown ales, themselves. You can have a Southern English Brown and a Northern English Brown and American Brown etc etc...
For example, here are the BJCP guidelines on US Amber and UK Brown ales:
10B. American Amber Ale
Aroma: Low to moderate hop aroma from dry hopping or late kettle additions of American hop varieties. A citrusy hop character is common, but not required. Moderately low to moderately high maltiness balances and sometimes masks the hop presentation, and usually shows a moderate caramel character. Esters vary from moderate to none. No diacetyl.
Appearance: Amber to coppery brown in color. Moderately large off-white head with good retention. Generally quite clear, although dry-hopped versions may be slightly hazy.
Flavor: Moderate to high hop flavor from American hop varieties, which often but not always has a citrusy quality. Malt flavors are moderate to strong, and usually show an initial malty sweetness followed by a moderate caramel flavor (and sometimes other character malts in lesser amounts). Malt and hop bitterness are usually balanced and mutually supportive. Fruity esters can be moderate to none. Caramel sweetness and hop flavor/bitterness can linger somewhat into the medium to full finish. No diacetyl.
Mouthfeel: Medium to medium-full body. Carbonation moderate to high. Overall smooth finish without astringency often associated with high hopping rates. Stronger versions may have a slight alcohol warmth.
Overall Impression: Like an American pale ale with more body, more caramel richness, and a balance more towards malt than hops (although hop rates can be significant).
History: Known simply as Red Ales in some regions, these beers were popularized in the hop-loving Northern California and the Pacific Northwest areas before spreading nationwide.
Comments: Can overlap in color with American pale ales. However, American amber ales differ from American pale ales not only by being usually darker in color, but also by having more caramel flavor, more body, and usually being balanced more evenly between malt and bitterness. Should not have a strong chocolate or roast character that might suggest an American brown ale (although small amounts are OK).
Ingredients: Pale ale malt, typically American two-row. Medium to dark crystal malts. May also contain specialty grains which add additional character and uniqueness. American hops, often with citrusy flavors, are common but others may also be used. Water can vary in sulfate and carbonate content.
Vital Statistics: OG: 1.045 – 1.060
IBUs: 25 – 40 FG: 1.010 – 1.015
SRM: 10 – 17 ABV: 4.5 – 6.2%
Commercial Examples: North Coast Red Seal Ale, Tröegs HopBack Amber Ale, Deschutes Cinder Cone Red, Pyramid Broken Rake, St. Rogue Red Ale, Anderson Valley Boont Amber Ale, Lagunitas Censored Ale, Avery Redpoint Ale, McNeill’s Firehouse Amber Ale, Mendocino Red Tail Ale, Bell's Amber
11C. Northern English Brown Ale
Aroma: Light, sweet malt aroma with toffee, nutty and/or caramel notes. A light but appealing fresh hop aroma (UK varieties) may also be noticed. A light fruity ester aroma may be evident in these beers, but should not dominate. Very low to no diacetyl.
Appearance: Dark amber to reddish-brown color. Clear. Low to moderate off-white to light tan head.
Flavor: Gentle to moderate malt sweetness, with a nutty, lightly caramelly character and a medium-dry to dry finish. Malt may also have a toasted, biscuity, or toffee-like character. Medium to medium-low bitterness. Malt-hop balance is nearly even, with hop flavor low to none (UK varieties). Some fruity esters can be present; low diacetyl (especially butterscotch) is optional but acceptable.
Mouthfeel: Medium-light to medium body. Medium to medium-high carbonation.
Overall Impression: Drier and more hop-oriented that southern English brown ale, with a nutty character rather than caramel.
History/Comments: English brown ales are generally split into sub-styles along geographic lines.
Ingredients: English mild ale or pale ale malt base with caramel malts. May also have small amounts darker malts (e.g., chocolate) to provide color and the nutty character. English hop varieties are most authentic. Moderate carbonate water.
Vital Statistics: OG: 1.040 – 1.052
IBUs: 20 – 30 FG: 1.008 – 1.013
SRM: 12 – 22 ABV: 4.2 – 5.4%
Commercial Examples: Newcastle Brown Ale, Samuel Smith’s Nut Brown Ale, Riggwelter Yorkshire Ale, Wychwood Hobgoblin, Tröegs Rugged Trail Ale, Alesmith Nautical Nut Brown Ale, Avery Ellie’s Brown Ale, Goose Island Nut Brown Ale, Samuel Adams Brown Ale
Cheers,
TL