Is appearance important at the border?

• There may be problems if your identity documents show no beard and you arrive with one. Immigration might believe that you are using the beard to disguise a stolen or forged passport. This problem should cause only delays if the biometrics check out.
• There may be problems whenever the immigration officer has discretion if a scruffy appearance is at odds with your stated purpose and financial situation. They are trained to look for inconsistencies and to question them. (For the UK, the immigration officer is not concerned with the purpose of your visit, only with your identity as an EU citizen. At least until Brexit.)

It shouldn't, but at least sometimes it does. Tony Saint worked in UK immigration for ten years, then wrote a novel based on his experiences. He called it Refusal Shoes, the title reflecting the propensity of some of his ex-colleagues for making admit/refuse decisions based solely on the applicant's footwear.

That said, as greatone notes, the UK (currently) has a treaty obligation to admit you unless it has a good reason not to, and "he was scruffy" is unlikely to fly as adequate.

Edit (in response to a comment): I'm fairly sure that EEA nationals have a right to enter unless any of several tightly-defined criteria are satisfied. Section 11 para 1 of the Immigration (EEA) Regulations 2006 says

An EEA national must be admitted to the United Kingdom if he produces on arrival a valid national identity card or passport issued by an EEA State.

So it's possible that the border officer may start asking questions - it's a free country, after all - but your admissibility doesn't depend on the answers to anything like the same degree they do for non-EEA nationals. The exceptions (ie, grounds for inadmissibility) are listed in s19 of the same regulations, if you're curious.

Disclaimer: I have no connection with Tony Saint or his book. I just own a copy, is all.

Your appearance should be consistent with your reason for coming. If you are here to "sightsee" but you look like a homeless person, wearing filthy clothes that don't fit well, carrying a broken piece of luggage tied up with rope, shoes with holes, etc, the officer is likely to think:

This person looks too poor and desperate to afford this vacation. Perhaps someone bought them a ticket in exchange for carrying something. Or perhaps they intend to take advantage of public services here and never leave. I'd better check a little more carefully.

If you are here to "sightsee" but you are wearing a 3 piece business suit, carrying a large briefcase, and wearing "slippery city shoes", the officer is likely to think:

This person might be here for a job interview or perhaps has a job already arranged. They could be planning to work without a permit or even to move here without the right paperwork. I'd better check a little more carefully.

If you are here to "visit a cousin" but you are wearing work boots, overalls, and a workshirt, the officer might think you don't own any non-work clothes, again leading to wondering how you can afford the trip and what's really the purpose of it, or they might think you intend to work.

Grooming issues, like how recently you shaved, factor into this a little: the person who is ungroomed to the point of society disapproving may trigger the "too poor to really be traveling" thoughts in an officer who should know better. And yes, there may be some officers who see a beard, especially on a middle eastern face, as a little threatening, and who subconsciously decide to check a little more carefully.

Now, in theory, this "check a little more carefully" shouldn't matter if all your papers are in order. You can look homeless, but if you can show you're gainfully employed and can easily afford the trip, the fact your shoes have holes in them shouldn't keep you out. You can look like an IT consultant headed to a client, but if you can show your itinerary and plans to tour the country by train and see this, this, and that museum, castle, lighthouse and lookout, then your business-like demeanor shouldn't keep you out. The same goes for looking somewhat threatening. In theory. Reality is not always like that, so some people work extra hard to make sure they don't trigger any extra checking or suspicion. It's your choice.

I cannot speak for Canada but if you arrive in the UK as an EU citizen with a biometric passport at one of the major airports you will be directed to the automatic gates. Unless the system which monitors the gates has some reason to reject you then you will never meet anyone from the Border Agency so your specific question about immigration officials answers itself. Of course you may be questioned by customs if they think you may have prohibited substances but that is not what you asked.

The UK is unlikely to ask anything except where you're flying from and possibly the purpose and duration of your stay. In general, they can only subject you to hassle if they suspect your passport is fake and you aren't really an EEA national.

Canada is a different beast. While not as "difficult" as the US, be prepared to convince them that you are a bona fide visitor and will return home. Appearance certainly helps, as it may be a telltale about your lifestyle in the eyes of officers. Furthermore they check more carefully that the passenger is the person on the passport.

As such, when going to Canada, if your beard makes you look nothing like your passport photo, shave it. Likewise, if anything categorically makes you look like something other than a regular visitor, adjust accordingly.

Of course your appearance makes a difference.

Note:
As seeing from the comments people do misunderstand what I try to say: You can not change your ethnicity, you can dress to make it more or less obvious, but the OP seems not to dress up to it now. So I leave ethnicity/race out of it, as it is a given. My answer is about how to give the best impression within the bound of ethnicity/race and chosen style of dress.

Immigration officers are human and all humans are influenced by how other humans look. They are less likely to be suspicious if you look like you combed your hair, washed your face and put on a fresh shirt.

That is not about what ethnicity you have (or look like you have) or whether your beard is suspicious in itself.
You can not change your ethnicity, so you should make sure you give the best impression you can.
The basic, 'person took time to clean up' against the 'same person looks like he slept in his clothes for weeks on end'. There is no need to travel in a business suit if you do not need a business suit where you go, just clean versions of your normal clothing should make the difference.

So take that few minutes and make sure you keep one clean shirt (or even better, whole outfit) to change into before you cross a border.

I even went to wearing thin white socks in my sandals when I was in my twenties, as to show as 'nice girl' rather than 'hippy', but only put them on after arriving on the airport and took them off when on the next transport.