Some thoughts from an Estonian's perspective.
It is certainly true that, being related languages, there is a remarkable similarity between Estonian and Finnish and many words coincide at least at the synonym level. However, I wouldn't be too optimistic about the mutual understandability when it comes to totally unprepared persons (i.e. those who have not learned or been otherwise actively exposed to the other language). Learning is certainly easier due to similarity (as the similarity also concerns the points that are classically considered difficult when learning either of the languages, such as word stem changes on declination, etc.) but on the other hand one has to be careful about it - besides the coinciding words there are words that are similar but have either shifted or completely different meanings. Some examples were already given on the former case ( like 'kiri' (est.'letter')/'kirja' (fin. 'book') let's add e.g. 'hoone' (est. 'building') / 'huone' (fin. 'room'), maybe also 'katsuma' (est. 'to touch, feel') / 'katsoa' (fin. 'to look')), but the latter case is especially awkward (e.g. 'ruumid' (est. 'rooms') / 'ruumit' (fin. 'corpses, dead bodies'), 'konn' (est. 'frog') / 'konna' (fin. 'bastard'), and one can find some even more drastic examples). The list is endless, in fact there exists an entire dictionary on such misinterpretations.
Another question is how many such 'unexposed' persons are there for example in Estonia these days. I think there may still be quite a lot. As to watching Finnish TV channels during the Soviet time, note that it was only possible in the Northern Estonia, basically just along the northern coast (which includes Tallinn, though). Naturally, in this area all kinds of influences of Finland and Finnish language have always been stronger than elsewhere, and one can find many people that speak Finnish there. And even though probably most people in Estonia have had some sort of connection with Finland or Finnish people by now, they often lack the motivation to do any language learning (especially having found out that it's not as easy as it may seem at first glance) and resort to English or some other language if needed.
How does the Finnish sound to Estonians? Well, it's a matter of taste, but I wouldn't call it 'mild' or 'childish'. Rather it sounds a bit glumsy and unevenly, due to generally longer words and the missing so-called 'second quantity' (middle-duration pronounciation of the stressed syllable) so all the words are either short or 'overlong'. There is less (if at all) palatalization in Finnish, which makes me even more puzzled about the 'mildness' someone could hear. I wouldn't say it's archaic either, although the words that are similar in both languages probably largely do represent the older part of the languages.
Some more remarks on the earlier postings:
Finnish: laskin (derived from "laskea", to count)
Estonian: kalkulaator
'Kalkulaator' is not very widely used, we tend to call everything that does some computing 'arvuti' (from 'arvutama', to calculate, compute), be it a desktop computer or a pocket calculator. You can say 'taskuarvuti' if you want to stress "pocketness" ('tasku'='pocket').
Another one, "spice"
Finnish: mauste (from "maustaa", to flavour)
Estonian: wvürts (German: Würze)
Does 'mauste' have to be spicy? I think the corresponding word in Estonian is 'maitseaine' ('maitse'='flavour', 'aine'='substance') which is just any kind of seasoning. The correspondence 'spice'='vürts' is probably correct, but to think about it, lately I've seen this word used more in various figurative contexts than talking about cooking.
And I just found this, "excavator"
Finnish: kaivinkone (from "kaivaa" to dig, and "kone" machine = digging machine)
Estonian: ekskavaator
That's right. Colloquially it's often called 'kopp', which actually means just the bucket (or the scoop, which may be where it's come from).
I'm not at all trying to prove here that the Estonian language is not into borrowing words. It certainly is, much more than Finnish that is one of the few languages having even a native word for 'sports' ('urheilu').